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.

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

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

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{ODEs}\right)\:$

$\mathrm{ode1}≔t^2\left(z\left(t\right)+1\right)+{z\left(t\right)}^2\left(t-1\right)\mathrm{diff}\left(z\left(t\right)\,t\right)=0$

$\mathrm{ode1}≔t^2\left(z\left(t\right)+1\right)+{z\left(t\right)}^2\left(t-1\right)\left(\frac{\ⅆ}{\ⅆt}z\left(t\right)\right)=0$

$\mathrm{Solve}\left(\mathrm{ode1}\,z\left(t\right)\right)$

$\frac{{z\left(t\right)}^2}{2}-z\left(t\right)+\ln \left(z\left(t\right)+1\right)=-\frac{t^2}{2}-t-\ln \left(t-1\right)+\mathrm{c\_\_1}$

$\mathrm{ic1}≔z\left(3\right)=1$

$\mathrm{ic1}≔z\left(3\right)=1$

$\mathrm{Solve}\left(\left\{\mathrm{ic1}\,\mathrm{ode1}\right\}\right)$

$\frac{{z\left(t\right)}^2}{2}-z\left(t\right)+\ln \left(z\left(t\right)+1\right)=-\frac{t^2}{2}-t-\ln \left(t-1\right)+7+2\ln \left(2\right)$

$\mathrm{ode2}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-\mathrm{diff}\left(y\left(x\right)\,x\right)-x\exp \left(x\right)=0$

$\mathrm{ode2}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)-\frac{\ⅆ}{\ⅆx}y\left(x\right)-x{\ⅇ}^x=0$

$\mathrm{Solve}\left(\mathrm{ode2}\,y\left(x\right)\right)$

$y\left(x\right)=\mathrm{c\_\_1}+\mathrm{c\_\_2}{\ⅇ}^x+{\ⅇ}^x\left(1-x+\frac{1}{2}{x}^2\right)$

$\mathrm{ode3}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+\frac{5{\mathrm{diff}\left(y\left(x\right)\,x\right)}^2}{y\left(x\right)}=0$

$\mathrm{ode3}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+\frac{5{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2}{y\left(x\right)}=0$

$\mathrm{Solve}\left(\mathrm{ode3}\,y\left(x\right)\right)$

$\left\{y\left(x\right)={\left(\mathrm{c\_\_1}x+\mathrm{c\_\_2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\,y\left(x\right)=\mathrm{c\_\_1}{\left(x+\mathrm{c\_\_2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\,y\left(x\right)=\mathrm{c\_\_2}{\left(\mathrm{c\_\_1}x+1\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}\right\}$

$\mathrm{ode4}≔x^3\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)+3x^2\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-6x\mathrm{diff}\left(y\left(x\right)\,x\right)-6y\left(x\right)=0$

$\mathrm{ode4}≔x^3\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)+3x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-6x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-6y\left(x\right)=0$

$\mathrm{Solve}\left(\mathrm{ode4}\,y\left(x\right)\right)$

$y\left(x\right)=\frac{\mathrm{c\_\_1}{x}^5+\mathrm{c\_\_3}x+\mathrm{c\_\_2}}{x^2}$

$\mathrm{sys5}≔\left\{\mathrm{diff}\left(y\left[1\right]\left(x\right)\,x\right)=7y\left[1\right]\left(x\right)+y\left[2\right]\left(x\right)\,\mathrm{diff}\left(y\left[2\right]\left(x\right)\,x\right)=-4y\left[1\right]\left(x\right)+3y\left[2\right]\left(x\right)\right\}$

$\mathrm{sys5}≔\left\{\frac{\ⅆ}{\ⅆx}{y}_1\left(x\right)=7y_1\left(x\right)+y_2\left(x\right)\,\frac{\ⅆ}{\ⅆx}{y}_2\left(x\right)=-4y_1\left(x\right)+3y_2\left(x\right)\right\}$

$\mathrm{Solve}\left(\mathrm{sys5}\right)$

$\left\{y_1\left(x\right)=-\frac{{\ⅇ}^{5x}\left(2\mathrm{c\_\_2}x+2\mathrm{c\_\_1}+\mathrm{c\_\_2}\right)}{4}\,y_2\left(x\right)={\ⅇ}^{5x}\left(\mathrm{c\_\_2}x+\mathrm{c\_\_1}\right)\right\}$

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

For more information on Maple 2021 changes, see Updates in Maple 2021.