The ByUndeterminedCoefficients(ODE, y(x)) command finds the solution of a linear ODE by the method of undetermined coefficients. This method is applicable when the coefficients of y(x) and its derivatives are constant and the forcing function is of a certain form, typically involving polynomials, exponentials, and trigonometric functions. This method works by determining the general form of a particular solution based on the form of the forcing function, substituting the proposed particular solution into the ODE, and solving for the undetermined 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.

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

$\mathrm{ode1}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+2\mathrm{diff}\left(y\left(x\right)\,x\right)+2y\left(x\right)=\sin \left(x\right)$

$\mathrm{ode1}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+2\frac{\ⅆ}{\ⅆx}y\left(x\right)+2y\left(x\right)=\sin \left(x\right)$

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

$y_p\left(x\right)=-\frac{2\cos \left(x\right)}{5}+\frac{\sin \left(x\right)}{5}$

$\mathrm{ode2}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+4\mathrm{diff}\left(y\left(x\right)\,x\right)+4y\left(x\right)=\exp \left(-2x\right)$

$\mathrm{ode2}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+4\frac{\ⅆ}{\ⅆx}y\left(x\right)+4y\left(x\right)={\ⅇ}^{-2x}$

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

$y_p\left(x\right)=\frac{x^2{\ⅇ}^{-2x}}{2}$

$\mathrm{ode3}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-4\mathrm{diff}\left(y\left(x\right)\,x\right)+4y\left(x\right)=x^2$

$\mathrm{ode3}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)-4\frac{\ⅆ}{\ⅆx}y\left(x\right)+4y\left(x\right)=x^2$

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

$y_p\left(x\right)=\frac{1}{4}{x}^2+\frac{1}{2}x+\frac{3}{8}$

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

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

The Student[ODEs][Solve][ByUndeterminedCoefficients] 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.