LinearConstantCoefficients - Maple Help
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Student[ODEs][Solve]

 LinearConstantCoefficients
 Solve a linear ODE with constant coefficients

 Calling Sequence LinearConstantCoefficients(ODE, y(x))

Parameters

 ODE - a linear ordinary differential equation with constant coefficients y - name; the dependent variable x - name; the independent variable

Description

 • The LinearConstantCoefficients(ODE, y(x)) command finds the solution of a linear ODE with constant coefficients.
 • 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

 > $\mathrm{with}\left({{\mathrm{Student}}_{\mathrm{ODEs}}}_{\mathrm{Solve}}\right):$
 > $\mathrm{ode1}≔\frac{ⅆ}{ⅆx}y\left(x\right)-6y\left(x\right)=0$
 ${\mathrm{ode1}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}{6}{}{y}{}\left({x}\right){=}{0}$ (1)
 > $\mathrm{LinearConstantCoefficients}\left(\mathrm{ode1},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{6}{}{x}}$ (2)
 > $\mathrm{ode2}≔\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)-\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)-6y\left(x\right)=0$
 ${\mathrm{ode2}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}{6}{}{y}{}\left({x}\right){=}{0}$ (3)
 > $\mathrm{LinearConstantCoefficients}\left(\mathrm{ode2},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{3}{}{x}}{+}{\mathrm{_C2}}{}{{ⅇ}}^{{-}{2}{}{x}}$ (4)
 > $\mathrm{ode3}≔\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)-\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)-6y\left(x\right)={x}^{2}$
 ${\mathrm{ode3}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}{6}{}{y}{}\left({x}\right){=}{{x}}^{{2}}$ (5)
 > $\mathrm{LinearConstantCoefficients}\left(\mathrm{ode3},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{3}{}{x}}{+}{\mathrm{_C2}}{}{{ⅇ}}^{{-}{2}{}{x}}{-}\frac{{{x}}^{{2}}}{{6}}{+}\frac{{x}}{{18}}{-}\frac{{7}}{{108}}$ (6)
 > $\mathrm{ode4}≔\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)+4y\left(x\right)+4\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)=0$
 ${\mathrm{ode4}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{4}{}{y}{}\left({x}\right){+}{4}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{0}$ (7)
 > $\mathrm{LinearConstantCoefficients}\left(\mathrm{ode4},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{-}{2}{}{x}}{+}{\mathrm{_C2}}{}{x}{}{{ⅇ}}^{{-}{2}{}{x}}$ (8)
 > $\mathrm{ode5}≔\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)+4y\left(x\right)+4\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)=-3\mathrm{sin}\left(x\right)$
 ${\mathrm{ode5}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{4}{}{y}{}\left({x}\right){+}{4}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{-}{3}{}{\mathrm{sin}}{}\left({x}\right)$ (9)
 > $\mathrm{LinearConstantCoefficients}\left(\mathrm{ode5},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{-}{2}{}{x}}{+}{\mathrm{_C2}}{}{x}{}{{ⅇ}}^{{-}{2}{}{x}}{+}\frac{{12}{}{\mathrm{cos}}{}\left({x}\right)}{{25}}{-}\frac{{9}{}{\mathrm{sin}}{}\left({x}\right)}{{25}}$ (10)
 > $\mathrm{ode6}≔\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)+2y\left(x\right)+2\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)=0$
 ${\mathrm{ode6}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{2}{}{y}{}\left({x}\right){+}{2}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{0}$ (11)
 > $\mathrm{LinearConstantCoefficients}\left(\mathrm{ode6},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{-}{x}}{}{\mathrm{sin}}{}\left({x}\right){+}{\mathrm{_C2}}{}{{ⅇ}}^{{-}{x}}{}{\mathrm{cos}}{}\left({x}\right)$ (12)
 > $\mathrm{ode7}≔\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)+2y\left(x\right)-2\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)={ⅇ}^{x}$
 ${\mathrm{ode7}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{2}{}{y}{}\left({x}\right){-}{2}{}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{{ⅇ}}^{{x}}$ (13)
 > $\mathrm{LinearConstantCoefficients}\left(\mathrm{ode7},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{x}}{}{\mathrm{sin}}{}\left({x}\right){+}{\mathrm{_C2}}{}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({x}\right){+}{{ⅇ}}^{{x}}$ (14)
 > $\mathrm{ode7}≔-\left(\frac{{ⅆ}^{3}}{ⅆ{x}^{3}}y\left(x\right)\right)+\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)-y\left(x\right)+\frac{ⅆ}{ⅆx}y\left(x\right)={ⅇ}^{x}$
 ${\mathrm{ode7}}{≔}{-}\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){-}{y}{}\left({x}\right){+}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{{ⅇ}}^{{x}}$ (15)
 > $\mathrm{LinearConstantCoefficients}\left(\mathrm{ode7},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{-}{x}}{+}{\mathrm{_C2}}{}{{ⅇ}}^{{x}}{+}{\mathrm{_C3}}{}{x}{}{{ⅇ}}^{{x}}{+}\frac{{{ⅇ}}^{{x}}{}\left({2}{}{{x}}^{{2}}{-}{2}{}{x}{+}{1}\right)}{{8}}$ (16)

Compatibility

 • The Student[ODEs][Solve][LinearConstantCoefficients] command was introduced in Maple 2021.
 • For more information on Maple 2021 changes, see Updates in Maple 2021.
 • The Student[ODEs][Solve][LinearConstantCoefficients] 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.

 See Also