FirstOrderLinear - Maple Help

Student[ODEs][Solve]

 FirstOrderLinear
 Solve a first order linear ODE

 Calling Sequence FirstOrderLinear(ODE, y(x))

Parameters

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

Description

 • 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.

Examples

 > $\mathrm{with}\left({{\mathrm{Student}}_{\mathrm{ODEs}}}_{\mathrm{Solve}}\right):$
 > $\mathrm{ode1}≔\frac{ⅆ}{ⅆt}x\left(t\right)+\mathrm{cos}\left(t\right)x\left(t\right)=1$
 ${\mathrm{ode1}}{≔}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}{\mathrm{cos}}{}\left({t}\right){}{x}{}\left({t}\right){=}{1}$ (1)
 > $\mathrm{FirstOrderLinear}\left(\mathrm{ode1},x\left(t\right)\right)$
 ${x}{}\left({t}\right){=}{{ⅇ}}^{{-}{\mathrm{sin}}{}\left({t}\right)}{}\left({\int }{{ⅇ}}^{{\mathrm{sin}}{}\left({t}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{t}{+}{\mathrm{_C1}}\right)$ (2)
 > $\mathrm{ode2}≔\frac{ⅆ}{ⅆt}x\left(t\right)-{ⅇ}^{t}x\left(t\right)=\mathrm{cos}\left(t\right)$
 ${\mathrm{ode2}}{≔}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){-}{{ⅇ}}^{{t}}{}{x}{}\left({t}\right){=}{\mathrm{cos}}{}\left({t}\right)$ (3)
 > $\mathrm{FirstOrderLinear}\left(\mathrm{ode2},x\left(t\right)\right)$
 ${x}{}\left({t}\right){=}{{ⅇ}}^{{{ⅇ}}^{{t}}}{}\left({\int }{{ⅇ}}^{{-}{{ⅇ}}^{{t}}}{}{\mathrm{cos}}{}\left({t}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{t}{+}{\mathrm{_C1}}\right)$ (4)
 > $\mathrm{ode3}≔\frac{ⅆ}{ⅆt}x\left(t\right)+x\left(t\right)-{t}^{2}=0$
 ${\mathrm{ode3}}{≔}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}{x}{}\left({t}\right){-}{{t}}^{{2}}{=}{0}$ (5)
 > $\mathrm{FirstOrderLinear}\left(\mathrm{ode3},x\left(t\right)\right)$
 ${x}{}\left({t}\right){=}{{t}}^{{2}}{-}{2}{}{t}{+}{2}{+}{\mathrm{_C1}}{}{{ⅇ}}^{{-}{t}}$ (6)
 > $\mathrm{ode4}≔\frac{ⅆ}{ⅆt}x\left(t\right)+{ⅇ}^{t}x\left(t\right)-{t}^{2}=0$
 ${\mathrm{ode4}}{≔}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}{{ⅇ}}^{{t}}{}{x}{}\left({t}\right){-}{{t}}^{{2}}{=}{0}$ (7)
 > $\mathrm{FirstOrderLinear}\left(\mathrm{ode4},x\left(t\right)\right)$
 ${x}{}\left({t}\right){=}{{ⅇ}}^{{-}{{ⅇ}}^{{t}}}{}\left({\int }{{ⅇ}}^{{{ⅇ}}^{{t}}}{}{{t}}^{{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{t}{+}{\mathrm{_C1}}\right)$ (8)
 > $\mathrm{ode5}≔\frac{ⅆ}{ⅆt}x\left(t\right)+tx\left(t\right)-\mathrm{sin}\left(t\right)=0$
 ${\mathrm{ode5}}{≔}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right){+}{t}{}{x}{}\left({t}\right){-}{\mathrm{sin}}{}\left({t}\right){=}{0}$ (9)
 > $\mathrm{FirstOrderLinear}\left(\mathrm{ode5},x\left(t\right)\right)$
 ${x}{}\left({t}\right){=}{-}\frac{\left(\sqrt{{\mathrm{\pi }}}{}{{ⅇ}}^{\frac{{1}}{{2}}}{}\sqrt{{2}}{}{\mathrm{erf}}{}\left(\frac{\sqrt{{2}}{}\left({I}{}{t}{-}{1}\right)}{{2}}\right){-}\sqrt{{\mathrm{\pi }}}{}{{ⅇ}}^{\frac{{1}}{{2}}}{}\sqrt{{2}}{}{\mathrm{erf}}{}\left(\frac{\sqrt{{2}}{}\left({I}{}{t}{+}{1}\right)}{{2}}\right){-}{4}{}{\mathrm{_C1}}\right){}{{ⅇ}}^{{-}\frac{{{t}}^{{2}}}{{2}}}}{{4}}$ (10)

Compatibility

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