The FirstOrderLinear(ODE, y(x)) command finds the solution of a first order linear ODE.

Use the option output=steps to make this command return an annotated step-by-step solution.  Further control over the format and display of the step-by-step solution is available using the options described in Student:-Basics:-OutputStepsRecord.  The options supported by that command can be passed to this one.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\left[\mathrm{Solve}\right]\right)\:$

$\mathrm{ode1}≔\mathrm{diff}\left(x\left(t\right)\,t\right)+\cos \left(t\right)x\left(t\right)=1$

$\mathrm{ode1}≔\frac{\ⅆ}{\ⅆt}x\left(t\right)+\cos \left(t\right)x\left(t\right)=1$

$\mathrm{FirstOrderLinear}\left(\mathrm{ode1}\,x\left(t\right)\right)$

$x\left(t\right)={\ⅇ}^{-\sin \left(t\right)}\left(\int {\ⅇ}^{\sin \left(t\right)}\ⅆt+\mathrm{\_C1}\right)$

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

$\mathrm{ode2}≔\frac{\ⅆ}{\ⅆt}x\left(t\right)-{\ⅇ}^{t}x\left(t\right)=\cos \left(t\right)$

$\mathrm{FirstOrderLinear}\left(\mathrm{ode2}\,x\left(t\right)\right)$

$x\left(t\right)={\ⅇ}^{{\ⅇ}^t}\left(\int {\ⅇ}^{-{\ⅇ}^t}\cos \left(t\right)\ⅆt+\mathrm{\_C1}\right)$

$\mathrm{ode3}≔\mathrm{diff}\left(x\left(t\right)\,t\right)+x\left(t\right)-t^2=0$

$\mathrm{ode3}≔\frac{\ⅆ}{\ⅆt}x\left(t\right)+x\left(t\right)-t^2=0$

$\mathrm{FirstOrderLinear}\left(\mathrm{ode3}\,x\left(t\right)\right)$

$x\left(t\right)=t^2-2t+2+\mathrm{\_C1}{\ⅇ}^{-t}$

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

$\mathrm{ode4}≔\frac{\ⅆ}{\ⅆt}x\left(t\right)+{\ⅇ}^{t}x\left(t\right)-t^2=0$

$\mathrm{FirstOrderLinear}\left(\mathrm{ode4}\,x\left(t\right)\right)$

$x\left(t\right)={\ⅇ}^{-{\ⅇ}^t}\left(\int {\ⅇ}^{{\ⅇ}^t}{t}^2\ⅆt+\mathrm{\_C1}\right)$

$\mathrm{ode5}≔\mathrm{diff}\left(x\left(t\right)\,t\right)+tx\left(t\right)-\sin \left(t\right)=0$

$\mathrm{ode5}≔\frac{\ⅆ}{\ⅆt}x\left(t\right)+tx\left(t\right)-\sin \left(t\right)=0$

$\mathrm{FirstOrderLinear}\left(\mathrm{ode5}\,x\left(t\right)\right)$

$x\left(t\right)=-\frac{\left(\sqrt{\mathrm{\pi }}{\ⅇ}^{\frac{1}{2}}\sqrt{2}\mathrm{erf}\left(\frac{\sqrt{2}\left(-1+\mathrm{I}t\right)}{2}\right)-\sqrt{\mathrm{\pi }}{\ⅇ}^{\frac{1}{2}}\sqrt{2}\mathrm{erf}\left(\frac{\sqrt{2}\left(1+\mathrm{I}t\right)}{2}\right)-4\mathrm{\_C1}\right){\ⅇ}^{-\frac{t^2}{2}}}{4}$

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

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

The Student[ODEs][Solve][FirstOrderLinear] command was updated in Maple 2022.

The output option was introduced in Maple 2022.

For more information on Maple 2022 changes, see Updates in Maple 2022.