SecondOrder - Maple Help

Student[ODEs][Solve]

 SecondOrder
 Solve a second order ODE

 Calling Sequence SecondOrder(ODE, y(x))

Parameters

 ODE - a second order ordinary differential equation y - name; the dependent variable x - name; the independent variable

Description

 • The SecondOrder(ODE, y(x)) command finds the solution of a second order ODE.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\left[\mathrm{Solve}\right]\right):$
 > $\mathrm{ode1}≔2x\mathrm{diff}\left(y\left(x\right),x\right)-9{x}^{2}+\left(2\mathrm{diff}\left(y\left(x\right),x\right)+{x}^{2}+1\right)\mathrm{diff}\left(y\left(x\right),x,x\right)=0$
 ${\mathrm{ode1}}{≔}{2}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{9}{}{{x}}^{{2}}{+}\left({2}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){=}{0}$ (1)
 > $\mathrm{SecondOrder}\left(\mathrm{ode1},y\left(x\right)\right)$
 $\left\{{y}{}\left({x}\right){=}{\int }\left({-}\frac{{{x}}^{{2}}}{{2}}{-}\frac{{1}}{{2}}{-}\frac{\sqrt{{{x}}^{{4}}{+}{12}{}{{x}}^{{3}}{+}{2}{}{{x}}^{{2}}{+}{4}{}{\mathrm{_C1}}{+}{1}}}{{2}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{+}{\mathrm{_C2}}{,}{y}{}\left({x}\right){=}{\int }\left({-}\frac{{{x}}^{{2}}}{{2}}{-}\frac{{1}}{{2}}{+}\frac{\sqrt{{{x}}^{{4}}{+}{12}{}{{x}}^{{3}}{+}{2}{}{{x}}^{{2}}{+}{4}{}{\mathrm{_C1}}{+}{1}}}{{2}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{+}{\mathrm{_C2}}\right\}$ (2)
 > $\mathrm{ode2}≔\mathrm{diff}\left(y\left(x\right),x,x\right)-\mathrm{diff}\left(y\left(x\right),x\right)-x\mathrm{exp}\left(x\right)=0$
 ${\mathrm{ode2}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}{x}{}{{ⅇ}}^{{x}}{=}{0}$ (3)
 > $\mathrm{SecondOrder}\left(\mathrm{ode2},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{+}{\mathrm{_C2}}{}{{ⅇ}}^{{x}}{+}\frac{{{ⅇ}}^{{x}}{}\left({{x}}^{{2}}{-}{2}{}{x}{+}{2}\right)}{{2}}$ (4)
 > $\mathrm{ode3}≔\mathrm{diff}\left(y\left(x\right),x,x\right)+\frac{5{\mathrm{diff}\left(y\left(x\right),x\right)}^{2}}{y\left(x\right)}=0$
 ${\mathrm{ode3}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}\frac{{5}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}^{{2}}}{{y}{}\left({x}\right)}{=}{0}$ (5)
 > $\mathrm{SecondOrder}\left(\mathrm{ode3},y\left(x\right)\right)$
 $\left\{{y}{}\left({x}\right){=}{\left({6}{}{{ⅇ}}^{{\mathrm{_C1}}}{}{x}{+}{6}{}{\mathrm{_C2}}\right)}^{{1}}{{6}}}{,}{y}{}\left({x}\right){=}{-}{\left({6}{}{{ⅇ}}^{{\mathrm{_C1}}}{}{x}{+}{6}{}{\mathrm{_C2}}\right)}^{{1}}{{6}}}\right\}$ (6)
 > $\mathrm{ode4}≔\mathrm{diff}\left(y\left(x\right),x,x\right)-\mathrm{diff}\left(y\left(x\right),x\right)-6y\left(x\right)=0$
 ${\mathrm{ode4}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}{6}{}{y}{}\left({x}\right){=}{0}$ (7)
 > $\mathrm{SecondOrder}\left(\mathrm{ode4},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{-}{2}{}{x}}{+}{\mathrm{_C2}}{}{{ⅇ}}^{{3}{}{x}}$ (8)
 > $\mathrm{ode5}≔\mathrm{diff}\left(y\left(x\right),x,x\right)-\mathrm{diff}\left(y\left(x\right),x\right)={x}^{2}+6y\left(x\right)$
 ${\mathrm{ode5}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{{x}}^{{2}}{+}{6}{}{y}{}\left({x}\right)$ (9)
 > $\mathrm{SecondOrder}\left(\mathrm{ode5},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{-}{2}{}{x}}{+}{\mathrm{_C2}}{}{{ⅇ}}^{{3}{}{x}}{-}\frac{{{x}}^{{2}}}{{6}}{+}\frac{{x}}{{18}}{-}\frac{{7}}{{108}}$ (10)
 > $\mathrm{ode6}≔\mathrm{diff}\left(y\left(x\right),x,x\right)+4y\left(x\right)=-4\mathrm{diff}\left(y\left(x\right),x\right)$
 ${\mathrm{ode6}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{4}{}{y}{}\left({x}\right){=}{-}{4}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)$ (11)
 > $\mathrm{SecondOrder}\left(\mathrm{ode6},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{-}{2}{}{x}}{+}{\mathrm{_C2}}{}{x}{}{{ⅇ}}^{{-}{2}{}{x}}$ (12)
 > $\mathrm{ode7}≔5\mathrm{diff}\left(y\left(x\right),x,x\right)+20y\left(x\right)+15\mathrm{sin}\left(x\right)=-20\mathrm{diff}\left(y\left(x\right),x\right)$
 ${\mathrm{ode7}}{≔}{5}{}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{20}{}{y}{}\left({x}\right){+}{15}{}{\mathrm{sin}}{}\left({x}\right){=}{-}{20}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)$ (13)
 > $\mathrm{SecondOrder}\left(\mathrm{ode7},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{-}{2}{}{x}}{+}{\mathrm{_C2}}{}{x}{}{{ⅇ}}^{{-}{2}{}{x}}{+}\frac{{12}{}{\mathrm{cos}}{}\left({x}\right)}{{25}}{-}\frac{{9}{}{\mathrm{sin}}{}\left({x}\right)}{{25}}$ (14)
 > $\mathrm{ode8}≔\mathrm{diff}\left(y\left(x\right),x,x\right)+2y\left(x\right)+2\mathrm{diff}\left(y\left(x\right),x\right)=0$
 ${\mathrm{ode8}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{2}{}{y}{}\left({x}\right){+}{2}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{0}$ (15)
 > $\mathrm{SecondOrder}\left(\mathrm{ode8},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{-}{x}}{}{\mathrm{cos}}{}\left({x}\right){+}{\mathrm{_C2}}{}{{ⅇ}}^{{-}{x}}{}{\mathrm{sin}}{}\left({x}\right)$ (16)
 > $\mathrm{ode9}≔\mathrm{diff}\left(y\left(x\right),x,x\right)+2y\left(x\right)-2\mathrm{diff}\left(y\left(x\right),x\right)=\mathrm{exp}\left(x\right)$
 ${\mathrm{ode9}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{2}{}{y}{}\left({x}\right){-}{2}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{{ⅇ}}^{{x}}$ (17)
 > $\mathrm{SecondOrder}\left(\mathrm{ode9},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({x}\right){+}{\mathrm{_C2}}{}{{ⅇ}}^{{x}}{}{\mathrm{sin}}{}\left({x}\right){+}{{ⅇ}}^{{x}}$ (18)

Compatibility

 • The Student[ODEs][Solve][SecondOrder] command was introduced in Maple 2021.