Question

Consider the differential equation $\mathit{a}\mathit{y}\mathbf{"}\mathbf{+}\mathit{b}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathit{c}\mathit{y}\mathbf{=}\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}$, where $\mathit{a}\mathbf{,}\mathit{b}\mathbf{,}$and$\mathit{c}$are constants. Choose the input functions$\mathit{g}\left(x\right)$for which the method of undetermined coefficients is applicable and the input functions for which the method of variation of parameters is applicable.

$\left(f\right)\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{x}}}{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{x}}$

Short answer

The method of undetermined coefficient is applied to$\left(f\right)g\left(x\right)=\frac{e^x}{\sin x}$

Step-by-step solution

Define 

In mathematics, the method of undetermined coefficients is an approach to finding a particular solution to certain nonhomogeneous ordinary differential equations and recurrence relations.

Solve the equation. 

For the second order differential equation$ay"+by\text{'}+cy=g\left(x\right)$:

The method of undetermined coefficients is applicable for

$\left(f\right)g\left(x\right)=\frac{e^x}{\sin x}$

For the given equation method of undetermined coefficients is applicable.