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.

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

$\mathrm{ode1}≔\mathrm{diff}\left(y\left(x\right)\,x\right)-6y\left(x\right)=0$

$\mathrm{ode1}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)-6y\left(x\right)=0$

$\mathrm{LinearConstantCoefficients}\left(\mathrm{ode1}\,y\left(x\right)\right)$

$y\left(x\right)=\mathrm{\_C1}{\ⅇ}^{6x}$

$\mathrm{ode2}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-\mathrm{diff}\left(y\left(x\right)\,x\right)-6y\left(x\right)=0$

$\mathrm{ode2}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)-\frac{\ⅆ}{\ⅆx}y\left(x\right)-6y\left(x\right)=0$

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

$y\left(x\right)=\mathrm{\_C1}{\ⅇ}^{3x}+\mathrm{\_C2}{\ⅇ}^{-2x}$

$\mathrm{ode3}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-\mathrm{diff}\left(y\left(x\right)\,x\right)-6y\left(x\right)=x^2$

$\mathrm{ode3}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)-\frac{\ⅆ}{\ⅆx}y\left(x\right)-6y\left(x\right)=x^2$

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

$y\left(x\right)=\mathrm{\_C1}{\ⅇ}^{3x}+\mathrm{\_C2}{\ⅇ}^{-2x}-\frac{x^2}{6}+\frac{x}{18}-\frac{7}{108}$

$\mathrm{ode4}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+4y\left(x\right)+4\mathrm{diff}\left(y\left(x\right)\,x\right)=0$

$\mathrm{ode4}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+4y\left(x\right)+4\frac{\ⅆ}{\ⅆx}y\left(x\right)=0$

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

$y\left(x\right)=\mathrm{\_C1}{\ⅇ}^{-2x}+\mathrm{\_C2}x{\ⅇ}^{-2x}$

$\mathrm{ode5}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+4y\left(x\right)+4\mathrm{diff}\left(y\left(x\right)\,x\right)=-3\sin \left(x\right)$

$\mathrm{ode5}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+4y\left(x\right)+4\frac{\ⅆ}{\ⅆx}y\left(x\right)=-3\sin \left(x\right)$

$\mathrm{LinearConstantCoefficients}\left(\mathrm{ode5}\,y\left(x\right)\right)$

$y\left(x\right)=\mathrm{\_C1}{\ⅇ}^{-2x}+\mathrm{\_C2}x{\ⅇ}^{-2x}+\frac{12\cos \left(x\right)}{25}-\frac{9\sin \left(x\right)}{25}$

$\mathrm{ode6}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+2y\left(x\right)+2\mathrm{diff}\left(y\left(x\right)\,x\right)=0$

$\mathrm{ode6}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+2y\left(x\right)+2\frac{\ⅆ}{\ⅆx}y\left(x\right)=0$

$\mathrm{LinearConstantCoefficients}\left(\mathrm{ode6}\,y\left(x\right)\right)$

$y\left(x\right)=\mathrm{\_C1}{\ⅇ}^{-x}\sin \left(x\right)+\mathrm{\_C2}{\ⅇ}^{-x}\cos \left(x\right)$

$\mathrm{ode7}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+2y\left(x\right)-2\mathrm{diff}\left(y\left(x\right)\,x\right)=\exp \left(x\right)$

$\mathrm{ode7}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+2y\left(x\right)-2\frac{\ⅆ}{\ⅆx}y\left(x\right)={\ⅇ}^x$

$\mathrm{LinearConstantCoefficients}\left(\mathrm{ode7}\,y\left(x\right)\right)$

$y\left(x\right)=\mathrm{\_C1}{\ⅇ}^x\sin \left(x\right)+\mathrm{\_C2}{\ⅇ}^x\cos \left(x\right)+{\ⅇ}^x$

$\mathrm{ode7}≔-\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)+\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-y\left(x\right)+\mathrm{diff}\left(y\left(x\right)\,x\right)=\exp \left(x\right)$

$\mathrm{ode7}≔-\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)+\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)-y\left(x\right)+\frac{\ⅆ}{\ⅆx}y\left(x\right)={\ⅇ}^x$

$\mathrm{LinearConstantCoefficients}\left(\mathrm{ode7}\,y\left(x\right)\right)$

$y\left(x\right)=\mathrm{\_C1}{\ⅇ}^{-x}+\mathrm{\_C2}{\ⅇ}^x+\mathrm{\_C3}x{\ⅇ}^x+\frac{{\ⅇ}^x\left(2x^2-2x+1\right)}{8}$

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.