Overview - Maple Help

Overview of the Student:-ODEs:-Solve Command and Subpackage

 Calling Sequence Student:-ODEs:-Solve:-command(arguments) command(arguments) Solve(ODE) Solve(ODE, vars)

Parameters

 ODE - an ordinary differential equation, an IVP, or a system vars - function, or a list or set of functions; the dependent variable(s), written in the form y(x)

Description

 • The Solve command finds the solution of an ordinary differential equation, an initial value problem, or a system of ODEs.
 • Using the option output=steps will cause this command to return an annotated step-by-step solution. To see what options are available to control the format and display the step-by-step solution, see Student:-Basics:-OutputStepsRecord. The options supported by that command can be passed into this one.
 • Student:-ODEs:-Solve is also a subpackage containing a number of commands for solving ordinary differential equations and systems of ODEs.
 • Each command in the Student:-ODEs:-Solve subpackage can be accessed by using either the long form or the short form of the command name in the command calling sequence.
 The long form, Student:-ODEs:-Solve:-command or Student:-ODEs:-Solve:-command, is always available. The short form can be used after loading the package.
 • The Maple Command Completion facility is helpful for entering the names of Student package commands.

Computation

 The subpackage Student:-ODEs:-Solve consists of commands for solving ODEs and systems according to various methods:

Getting Help with a Command in the Package

 To display the help page for a particular Student:-ODEs:-Solve command, see Getting Help with a Command in a Package.

Examples

 > $\mathrm{with}\left(\mathrm{Student}:-\mathrm{ODEs}\right):$
 > $\mathrm{ode1}≔{t}^{2}\left(z\left(t\right)+1\right)+{z\left(t\right)}^{2}\left(t-1\right)\mathrm{diff}\left(z\left(t\right),t\right)=0$
 ${\mathrm{ode1}}{≔}{{t}}^{{2}}{}\left({z}{}\left({t}\right){+}{1}\right){+}{{z}{}\left({t}\right)}^{{2}}{}\left({t}{-}{1}\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({t}\right)\right){=}{0}$ (1)
 > $\mathrm{Solve}\left(\mathrm{ode1},z\left(t\right)\right)$
 $\frac{{{z}{}\left({t}\right)}^{{2}}}{{2}}{-}{z}{}\left({t}\right){+}{\mathrm{ln}}{}\left({z}{}\left({t}\right){+}{1}\right){=}{-}\frac{{{t}}^{{2}}}{{2}}{-}{t}{-}{\mathrm{ln}}{}\left({t}{-}{1}\right){+}{\mathrm{_C1}}$ (2)
 > $\mathrm{ic1}≔z\left(3\right)=1$
 ${\mathrm{ic1}}{≔}{z}{}\left({3}\right){=}{1}$ (3)
 > $\mathrm{Solve}\left(\left\{\mathrm{ic1},\mathrm{ode1}\right\}\right)$
 $\frac{{{z}{}\left({t}\right)}^{{2}}}{{2}}{-}{z}{}\left({t}\right){+}{\mathrm{ln}}{}\left({z}{}\left({t}\right){+}{1}\right){=}{-}\frac{{{t}}^{{2}}}{{2}}{-}{t}{-}{\mathrm{ln}}{}\left({t}{-}{1}\right){+}{7}{+}{2}{}{\mathrm{ln}}{}\left({2}\right)$ (4)
 > $\mathrm{ode2}≔\mathrm{diff}\left(y\left(x\right),x,x\right)-\mathrm{diff}\left(y\left(x\right),x\right)-x\mathrm{exp}\left(x\right)=0$
 ${\mathrm{ode2}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}{x}{}{{ⅇ}}^{{x}}{=}{0}$ (5)
 > $\mathrm{Solve}\left(\mathrm{ode2},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{+}{\mathrm{_C2}}{}{{ⅇ}}^{{x}}{+}{{ⅇ}}^{{x}}{}\left({1}{-}{x}{+}\frac{{1}}{{2}}{}{{x}}^{{2}}\right)$ (6)
 > $\mathrm{ode3}≔\mathrm{diff}\left(y\left(x\right),x,x\right)+\frac{5{\mathrm{diff}\left(y\left(x\right),x\right)}^{2}}{y\left(x\right)}=0$
 ${\mathrm{ode3}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}\frac{{5}{}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}^{{2}}}{{y}{}\left({x}\right)}{=}{0}$ (7)
 > $\mathrm{Solve}\left(\mathrm{ode3},y\left(x\right)\right)$
 $\left\{{y}{}\left({x}\right){=}{\left({6}{}{{ⅇ}}^{\mathrm{c__1}}{}{x}{+}{6}{}{\mathrm{_C2}}\right)}^{{1}}{{6}}}{,}{y}{}\left({x}\right){=}{-}{\left({6}{}{{ⅇ}}^{\mathrm{c__1}}{}{x}{+}{6}{}{\mathrm{_C2}}\right)}^{{1}}{{6}}}\right\}$ (8)
 > $\mathrm{ode4}≔{x}^{3}\mathrm{diff}\left(y\left(x\right),x,x,x\right)+3{x}^{2}\mathrm{diff}\left(y\left(x\right),x,x\right)-6x\mathrm{diff}\left(y\left(x\right),x\right)-6y\left(x\right)=0$
 ${\mathrm{ode4}}{≔}{{x}}^{{3}}{}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{3}{}{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{6}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{6}{}{y}{}\left({x}\right){=}{0}$ (9)
 > $\mathrm{Solve}\left(\mathrm{ode4},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}\frac{{4}{}{\mathrm{_C3}}{}{{x}}^{{5}}{+}{36}{}{\mathrm{_C2}}{}{x}{+}{9}{}\mathrm{c__1}}{{36}{}{{x}}^{{2}}}$ (10)
 > $\mathrm{sys5}≔\left\{\mathrm{diff}\left(y\left[1\right]\left(x\right),x\right)=7y\left[1\right]\left(x\right)+y\left[2\right]\left(x\right),\mathrm{diff}\left(y\left[2\right]\left(x\right),x\right)=-4y\left[1\right]\left(x\right)+3y\left[2\right]\left(x\right)\right\}$
 ${\mathrm{sys5}}{≔}\left\{\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{1}}{}\left({x}\right){=}{7}{}{{y}}_{{1}}{}\left({x}\right){+}{{y}}_{{2}}{}\left({x}\right){,}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{y}}_{{2}}{}\left({x}\right){=}{-}{4}{}{{y}}_{{1}}{}\left({x}\right){+}{3}{}{{y}}_{{2}}{}\left({x}\right)\right\}$ (11)
 > $\mathrm{Solve}\left(\mathrm{sys5}\right)$
 $\left\{{{y}}_{{1}}{}\left({x}\right){=}{-}\frac{{{ⅇ}}^{{5}{}{x}}{}\left({2}{}{\mathrm{_C2}}{}{x}{+}{\mathrm{_C2}}{+}{2}{}\mathrm{c__1}\right)}{{4}}{,}{{y}}_{{2}}{}\left({x}\right){=}{{ⅇ}}^{{5}{}{x}}{}\left({\mathrm{_C2}}{}{x}{+}\mathrm{c__1}\right)\right\}$ (12)

Compatibility

 • The Student:-ODEs:-Solve package was introduced in Maple 2021.