Special Function Solutions - Maple Help

ODE Steps for Special Function Solutions

Overview

 • This help page gives a few examples of using the command ODESteps to solve ordinary differential equations in terms of special functions.
 • See Student[ODEs][ODESteps] for a general description of the command ODESteps and its calling sequence.

Examples

 > $\mathrm{with}\left(\mathrm{Student}:-\mathrm{ODEs}\right):$
 > $\mathrm{ode1}≔{x}^{2}\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)+4x\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+\left(25{x}^{2}-9\right)y\left(x\right)=0$
 ${\mathrm{ode1}}{≔}{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{4}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({25}{}{{x}}^{{2}}{-}{9}\right){}{y}{}\left({x}\right){=}{0}$ (1)
 > $\mathrm{ODESteps}\left(\mathrm{ode1}\right)$
 $\begin{array}{lll}{}& {}& \text{Let's solve}\\ {}& {}& {{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{4}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({25}{}{{x}}^{{2}}{-}{9}\right){}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{Highest derivative means the order of the ODE is}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}2\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\\ \text{•}& {}& \text{Isolate 2nd derivative}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{-}\frac{\left({25}{}{{x}}^{{2}}{-}{9}\right){}{y}{}\left({x}\right)}{{{x}}^{{2}}}{-}\frac{{4}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{{x}}\\ \text{•}& {}& \text{Group terms with}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}y{}\left(x\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}\frac{{4}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{{x}}{+}\frac{\left({25}{}{{x}}^{{2}}{-}{9}\right){}{y}{}\left({x}\right)}{{{x}}^{{2}}}{=}{0}\\ \text{•}& {}& \text{Simplify ODE}\\ {}& {}& {{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{25}{}{y}{}\left({x}\right){}{{x}}^{{2}}{+}{4}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{9}{}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{Make a change of variables}\\ {}& {}& {t}{=}{5}{}{x}\\ \text{•}& {}& \text{Compute}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{ⅆ}{ⅆx}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y{}\left(x\right)\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{5}{}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\\ \text{•}& {}& \text{Compute second derivative}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{25}{}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\\ \text{•}& {}& \text{Apply change of variables to the ODE}\\ {}& {}& {{t}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}{y}{}\left({t}\right){}{{t}}^{{2}}{+}{4}{}{t}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){-}{9}{}{y}{}\left({t}\right){=}{0}\\ \text{•}& {}& \text{Make a change of variables}\\ {}& {}& {y}{}\left({t}\right){=}\frac{{u}{}\left({t}\right)}{{{t}}^{{3}}{{2}}}}\\ \text{•}& {}& \text{Compute}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{ⅆ}{ⅆt}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y{}\left(t\right)\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){=}{-}\frac{{3}{}{u}{}\left({t}\right)}{{2}{}{{t}}^{{5}}{{2}}}}{+}\frac{\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}\right)}{{{t}}^{{3}}{{2}}}}\\ \text{•}& {}& \text{Compute}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{{ⅆ}^{2}}{ⅆ{t}^{2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y{}\left(t\right)\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){=}\frac{{15}{}{u}{}\left({t}\right)}{{4}{}{{t}}^{{7}}{{2}}}}{-}\frac{{3}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}\right)\right)}{{{t}}^{{5}}{{2}}}}{+}\frac{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}\right)}{{{t}}^{{3}}{{2}}}}\\ \text{•}& {}& \text{Apply change of variables to the ODE}\\ {}& {}& {u}{}\left({t}\right){}{{t}}^{{2}}{+}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}\right)\right){}{{t}}^{{2}}{+}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}\right)\right){}{t}{-}\frac{{45}{}{u}{}\left({t}\right)}{{4}}{=}{0}\\ \text{•}& {}& \text{ODE is now of the Bessel form}\\ \text{•}& {}& \text{Solution to Bessel ODE}\\ {}& {}& {u}{}\left({t}\right){=}{\mathrm{C1}}{}{\mathrm{BesselJ}}{}\left(\frac{{3}{}\sqrt{{5}}}{{2}}{,}{t}\right){+}{\mathrm{C2}}{}{\mathrm{BesselY}}{}\left(\frac{{3}{}\sqrt{{5}}}{{2}}{,}{t}\right)\\ \text{•}& {}& \text{Make the change from}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}y{}\left(x\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{back to}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}y{}\left(t\right)\\ {}& {}& {y}{}\left({t}\right){=}\frac{{\mathrm{C1}}{}{\mathrm{BesselJ}}{}\left(\frac{{3}{}\sqrt{{5}}}{{2}}{,}{t}\right){+}{\mathrm{C2}}{}{\mathrm{BesselY}}{}\left(\frac{{3}{}\sqrt{{5}}}{{2}}{,}{t}\right)}{{{t}}^{{3}}{{2}}}}\\ \text{•}& {}& \text{Make the change from}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}t\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{back to}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}x\\ {}& {}& {y}{}\left({x}\right){=}\frac{\left({\mathrm{C1}}{}{\mathrm{BesselJ}}{}\left(\frac{{3}{}\sqrt{{5}}}{{2}}{,}{5}{}{x}\right){+}{\mathrm{C2}}{}{\mathrm{BesselY}}{}\left(\frac{{3}{}\sqrt{{5}}}{{2}}{,}{5}{}{x}\right)\right){}\sqrt{{5}}}{{25}{}{{x}}^{{3}}{{2}}}}\end{array}$ (2)
 > $\mathrm{ode2}≔\left(-{x}^{2}+1\right)\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)-x\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+y\left(x\right)=0$
 ${\mathrm{ode2}}{≔}\left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{y}{}\left({x}\right){=}{0}$ (3)
 > $\mathrm{ODESteps}\left(\mathrm{ode2}\right)$
 $\begin{array}{lll}{}& {}& \text{Let's solve}\\ {}& {}& \left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{Highest derivative means the order of the ODE is}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}2\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\\ \text{•}& {}& \text{Isolate 2nd derivative}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\frac{{y}{}\left({x}\right)}{{{x}}^{{2}}{-}{1}}{-}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{{{x}}^{{2}}{-}{1}}\\ \text{•}& {}& \text{Group terms with}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}y{}\left(x\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{{{x}}^{{2}}{-}{1}}{-}\frac{{y}{}\left({x}\right)}{{{x}}^{{2}}{-}{1}}{=}{0}\\ \text{•}& {}& \text{Multiply by denominators of ODE}\\ {}& {}& \left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{Make a change of variables}\\ {}& {}& {\mathrm{\theta }}{=}{\mathrm{arccos}}{}\left({x}\right)\\ \text{•}& {}& \text{Calculate}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{ⅆ}{ⅆx}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y{}\left(x\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{with change of variables}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\theta }}{}\left({x}\right)\right)\\ \text{•}& {}& \text{Compute}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{1st}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{derivative}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{ⅆ}{ⅆx}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y{}\left(x\right)\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{-}\frac{\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)}{\sqrt{{-}{{x}}^{{2}}{+}{1}}}\\ \text{•}& {}& \text{Calculate}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y{}\left(x\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{with change of variables}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right){}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\theta }}{}\left({x}\right)\right)}^{{2}}{+}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\theta }}{}\left({x}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)\\ \text{•}& {}& \text{Compute}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{2nd}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{derivative}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y{}\left(x\right)\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\frac{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)}{{-}{{x}}^{{2}}{+}{1}}{-}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}\\ \text{•}& {}& \text{Apply the change of variables to the ODE}\\ {}& {}& \left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)}{{-}{{x}}^{{2}}{+}{1}}{-}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}\right){+}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{\sqrt{{-}{{x}}^{{2}}{+}{1}}}{+}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{Multiply through}\\ {}& {}& {-}\frac{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right){}{{x}}^{{2}}}{{-}{{x}}^{{2}}{+}{1}}{+}\frac{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)}{{-}{{x}}^{{2}}{+}{1}}{+}\frac{{{x}}^{{3}}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}{-}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}{+}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{\sqrt{{-}{{x}}^{{2}}{+}{1}}}{+}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{Simplify ODE}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right){+}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{ODE is that of a harmonic oscillator with given general solution}\\ {}& {}& {y}{}\left({\mathrm{\theta }}\right){=}{\mathrm{C1}}{}\left[{}\right]{+}{\mathrm{C2}}{}\left[{}\right]\\ \text{•}& {}& \text{Revert back to}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}x\\ {}& {}& {y}{}\left({x}\right){=}{\mathrm{C1}}{}\left[{}\right]{+}{\mathrm{C2}}{}\left[{}\right]\\ \text{•}& {}& \text{Use trig identity to simplify}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\mathrm{sin}{}\left(\mathrm{arccos}{}\left(x\right)\right)\\ {}& {}& \left[{}\right]{=}\sqrt{{-}{{x}}^{{2}}{+}{1}}\\ \text{•}& {}& \text{Simplify solution to the ODE}\\ {}& {}& {y}{}\left({x}\right){=}{\mathrm{C1}}{}\sqrt{{-}{{x}}^{{2}}{+}{1}}{+}{\mathrm{C2}}{}{x}\end{array}$ (4)
 > $\mathrm{ode3}≔\left(-{x}^{2}+1\right)\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)-x\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+4y\left(x\right)=0$
 ${\mathrm{ode3}}{≔}\left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{4}{}{y}{}\left({x}\right){=}{0}$ (5)
 > $\mathrm{ODESteps}\left(\mathrm{ode3}\right)$
 $\begin{array}{lll}{}& {}& \text{Let's solve}\\ {}& {}& \left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{4}{}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{Highest derivative means the order of the ODE is}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}2\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\\ \text{•}& {}& \text{Isolate 2nd derivative}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\frac{{4}{}{y}{}\left({x}\right)}{{{x}}^{{2}}{-}{1}}{-}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{{{x}}^{{2}}{-}{1}}\\ \text{•}& {}& \text{Group terms with}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}y{}\left(x\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{{{x}}^{{2}}{-}{1}}{-}\frac{{4}{}{y}{}\left({x}\right)}{{{x}}^{{2}}{-}{1}}{=}{0}\\ \text{•}& {}& \text{Multiply by denominators of ODE}\\ {}& {}& \left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{4}{}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{Make a change of variables}\\ {}& {}& {\mathrm{\theta }}{=}{\mathrm{arccos}}{}\left({x}\right)\\ \text{•}& {}& \text{Calculate}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{ⅆ}{ⅆx}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y{}\left(x\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{with change of variables}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\theta }}{}\left({x}\right)\right)\\ \text{•}& {}& \text{Compute}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{1st}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{derivative}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{ⅆ}{ⅆx}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y{}\left(x\right)\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{-}\frac{\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)}{\sqrt{{-}{{x}}^{{2}}{+}{1}}}\\ \text{•}& {}& \text{Calculate}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y{}\left(x\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{with change of variables}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right){}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\theta }}{}\left({x}\right)\right)}^{{2}}{+}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\theta }}{}\left({x}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)\\ \text{•}& {}& \text{Compute}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{2nd}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{derivative}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y{}\left(x\right)\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\frac{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)}{{-}{{x}}^{{2}}{+}{1}}{-}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}\\ \text{•}& {}& \text{Apply the change of variables to the ODE}\\ {}& {}& \left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)}{{-}{{x}}^{{2}}{+}{1}}{-}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}\right){+}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{\sqrt{{-}{{x}}^{{2}}{+}{1}}}{+}{4}{}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{Multiply through}\\ {}& {}& {-}\frac{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right){}{{x}}^{{2}}}{{-}{{x}}^{{2}}{+}{1}}{+}\frac{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)}{{-}{{x}}^{{2}}{+}{1}}{+}\frac{{{x}}^{{3}}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}{-}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}{+}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{\sqrt{{-}{{x}}^{{2}}{+}{1}}}{+}{4}{}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{Simplify ODE}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right){+}{4}{}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{ODE is that of a harmonic oscillator with given general solution}\\ {}& {}& {y}{}\left({\mathrm{\theta }}\right){=}{\mathrm{C1}}{}\left[{}\right]{+}{\mathrm{C2}}{}\left[{}\right]\\ \text{•}& {}& \text{Revert back to}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}x\\ {}& {}& {y}{}\left({x}\right){=}{\mathrm{C1}}{}\left[{}\right]{+}{\mathrm{C2}}{}\left[{}\right]\\ \text{•}& {}& \text{Apply double angle identities to solution}\\ {}& {}& {y}{}\left({x}\right){=}{\mathrm{C1}}{}\left[{}\right]{}\left[{}\right]{+}{\mathrm{C2}}{}\left({2}{}{\left[{}\right]}^{{2}}{-}{1}\right)\\ \text{•}& {}& \text{Use trig identity to simplify sin}\\ {}& {}& \left[{}\right]{=}\sqrt{{-}{{x}}^{{2}}{+}{1}}\\ \text{•}& {}& \text{Simplify solution to the ODE}\\ {}& {}& {y}{}\left({x}\right){=}{\mathrm{C1}}{}{x}{}\sqrt{{-}{{x}}^{{2}}{+}{1}}{+}{\mathrm{C2}}{}\left({2}{}{{x}}^{{2}}{-}{1}\right)\end{array}$ (6)
 > $\mathrm{ode4}≔\left(-{x}^{2}+1\right)\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)-x\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)+9y\left(x\right)=0$
 ${\mathrm{ode4}}{≔}\left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{9}{}{y}{}\left({x}\right){=}{0}$ (7)
 > $\mathrm{ODESteps}\left(\mathrm{ode4}\right)$
 $\begin{array}{lll}{}& {}& \text{Let's solve}\\ {}& {}& \left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{9}{}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{Highest derivative means the order of the ODE is}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}2\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\\ \text{•}& {}& \text{Isolate 2nd derivative}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\frac{{9}{}{y}{}\left({x}\right)}{{{x}}^{{2}}{-}{1}}{-}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{{{x}}^{{2}}{-}{1}}\\ \text{•}& {}& \text{Group terms with}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}y{}\left(x\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{{{x}}^{{2}}{-}{1}}{-}\frac{{9}{}{y}{}\left({x}\right)}{{{x}}^{{2}}{-}{1}}{=}{0}\\ \text{•}& {}& \text{Multiply by denominators of ODE}\\ {}& {}& \left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{9}{}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{Make a change of variables}\\ {}& {}& {\mathrm{\theta }}{=}{\mathrm{arccos}}{}\left({x}\right)\\ \text{•}& {}& \text{Calculate}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{ⅆ}{ⅆx}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y{}\left(x\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{with change of variables}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\theta }}{}\left({x}\right)\right)\\ \text{•}& {}& \text{Compute}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{1st}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{derivative}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{ⅆ}{ⅆx}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y{}\left(x\right)\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{-}\frac{\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)}{\sqrt{{-}{{x}}^{{2}}{+}{1}}}\\ \text{•}& {}& \text{Calculate}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y{}\left(x\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{with change of variables}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right){}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\theta }}{}\left({x}\right)\right)}^{{2}}{+}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\theta }}{}\left({x}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)\\ \text{•}& {}& \text{Compute}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{2nd}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{derivative}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y{}\left(x\right)\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\frac{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)}{{-}{{x}}^{{2}}{+}{1}}{-}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}\\ \text{•}& {}& \text{Apply the change of variables to the ODE}\\ {}& {}& \left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)}{{-}{{x}}^{{2}}{+}{1}}{-}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}\right){+}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{\sqrt{{-}{{x}}^{{2}}{+}{1}}}{+}{9}{}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{Multiply through}\\ {}& {}& {-}\frac{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right){}{{x}}^{{2}}}{{-}{{x}}^{{2}}{+}{1}}{+}\frac{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)}{{-}{{x}}^{{2}}{+}{1}}{+}\frac{{{x}}^{{3}}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}{-}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}{+}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{\sqrt{{-}{{x}}^{{2}}{+}{1}}}{+}{9}{}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{Simplify ODE}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right){+}{9}{}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{ODE is that of a harmonic oscillator with given general solution}\\ {}& {}& {y}{}\left({\mathrm{\theta }}\right){=}{\mathrm{C1}}{}\left[{}\right]{+}{\mathrm{C2}}{}\left[{}\right]\\ \text{•}& {}& \text{Revert back to}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}x\\ {}& {}& {y}{}\left({x}\right){=}{\mathrm{C1}}{}\left[{}\right]{+}{\mathrm{C2}}{}\left[{}\right]\end{array}$ (8)
 > $\mathrm{ode5}≔\left(-{x}^{2}+1\right)\left(\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)-x\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)-4y\left(x\right)=0$
 ${\mathrm{ode5}}{≔}\left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{4}{}{y}{}\left({x}\right){=}{0}$ (9)
 > $\mathrm{ODESteps}\left(\mathrm{ode5}\right)$
 $\begin{array}{lll}{}& {}& \text{Let's solve}\\ {}& {}& \left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{4}{}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{Highest derivative means the order of the ODE is}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}2\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\\ \text{•}& {}& \text{Isolate 2nd derivative}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{-}\frac{{4}{}{y}{}\left({x}\right)}{{{x}}^{{2}}{-}{1}}{-}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{{{x}}^{{2}}{-}{1}}\\ \text{•}& {}& \text{Group terms with}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}y{}\left(x\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{{{x}}^{{2}}{-}{1}}{+}\frac{{4}{}{y}{}\left({x}\right)}{{{x}}^{{2}}{-}{1}}{=}{0}\\ \text{•}& {}& \text{Multiply by denominators of ODE}\\ {}& {}& \left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{4}{}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{Make a change of variables}\\ {}& {}& {\mathrm{\theta }}{=}{\mathrm{arccos}}{}\left({x}\right)\\ \text{•}& {}& \text{Calculate}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{ⅆ}{ⅆx}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y{}\left(x\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{with change of variables}\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\theta }}{}\left({x}\right)\right)\\ \text{•}& {}& \text{Compute}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{1st}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{derivative}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{ⅆ}{ⅆx}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y{}\left(x\right)\\ {}& {}& \frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{-}\frac{\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)}{\sqrt{{-}{{x}}^{{2}}{+}{1}}}\\ \text{•}& {}& \text{Calculate}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y{}\left(x\right)\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{with change of variables}\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right){}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\theta }}{}\left({x}\right)\right)}^{{2}}{+}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\theta }}{}\left({x}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)\\ \text{•}& {}& \text{Compute}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{2nd}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{derivative}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y{}\left(x\right)\\ {}& {}& \frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\frac{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)}{{-}{{x}}^{{2}}{+}{1}}{-}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}\\ \text{•}& {}& \text{Apply the change of variables to the ODE}\\ {}& {}& \left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)}{{-}{{x}}^{{2}}{+}{1}}{-}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}\right){+}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{\sqrt{{-}{{x}}^{{2}}{+}{1}}}{-}{4}{}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{Multiply through}\\ {}& {}& {-}\frac{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right){}{{x}}^{{2}}}{{-}{{x}}^{{2}}{+}{1}}{+}\frac{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)}{{-}{{x}}^{{2}}{+}{1}}{+}\frac{{{x}}^{{3}}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}{-}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{{\left({-}{{x}}^{{2}}{+}{1}\right)}^{{3}}{{2}}}}{+}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right)\right)}{\sqrt{{-}{{x}}^{{2}}{+}{1}}}{-}{4}{}{y}{}\left({x}\right){=}{0}\\ \text{•}& {}& \text{Simplify ODE}\\ {}& {}& {-}{4}{}{y}{}\left({x}\right){+}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{\theta }}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({\mathrm{\theta }}\right){=}{0}\\ \text{•}& {}& \text{ODE is second order linear with characteristic polynomial that is the difference of squares with given general solution}\\ {}& {}& {y}{}\left({\mathrm{\theta }}\right){=}{\mathrm{C1}}{}{{ⅇ}}^{{2}{}{\mathrm{\theta }}}{+}{\mathrm{C2}}{}{{ⅇ}}^{{-}{2}{}{\mathrm{\theta }}}\\ \text{•}& {}& \text{Revert back to}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}x\\ {}& {}& {y}{}\left({x}\right){=}{\mathrm{C1}}{}{{ⅇ}}^{{2}{}{\mathrm{arccos}}{}\left({x}\right)}{+}{\mathrm{C2}}{}{{ⅇ}}^{{-}{2}{}{\mathrm{arccos}}{}\left({x}\right)}\end{array}$ (10)