Question

Using Problems 29 and 31b show that equation (6.24) is correct.

Short answer

Answer

It is proved that $y_p=\left\{\begin{array}{l}{e}^{cx}{Q}_n\left(x\right)ifc\ne toeitheraorb\\ x{e}^{cx}{Q}_n\left(x\right)ifc=toeitheraorb,a\ne b\\ x^2{e}^{cx}{Q}_n\left(x\right)ifc=a=b\end{array}\right.$

Step-by-step solution

Given data.

Information given in the question is $\left(D-a\right)\left(D-b\right)y=\left\{\begin{array}{c}k\sin ax\\ \\ k\cos ax\end{array}\right\}$

Condition to prove the statement.

Here $y_p$is a polynomial ${\mathit{Q}}_{\mathbf{n}}\left(x\right)$of degree $\mathit{n}$

${\mathit{y}}_{\mathbf{p}}\mathbf{=}\{\begin{array}{l}{e}^{cx}{Q}_n\left(x\right)ifc\ne toeitheraorb\\ x{e}^{cx}{Q}_n\left(x\right)ifc=toeitheraorb,a\ne b\\ x^2{e}^{cx}{Q}_n\left(x\right)ifc=a=b\end{array}$

Find the particular solution of (D-a)(D-b)y={k sin axk cos ax}

A particular solution of $\left(D-a\right)\left(D-b\right)y=\left\{\begin{array}{c}k\sin ax\\ \\ k\cos ax\end{array}\right\}$

First solve

$$\begin{gathered} \left(D-a\right)\left(D-b\right)\mathit{y}\mathbf{=}\mathit{k}{\mathit{e}}^{\mathbf{k}\mathbf{i}\mathbf{x}} \\ \left(D-a\right)\left(D-b\right)\mathit{y}\mathbf{=}\mathit{k}\left(cosax+ininax\right) \\ \left(D-a\right)\left(D-b\right)\mathit{y}\mathbf{=}\left(kcosax+Kininax\right) \end{gathered}$$

Take the real and imaginary part

$\left(D-a\right)\left(D-b\right)y=\left\{\begin{array}{c}k\sin ax\\ \\ k\cos ax\end{array}\right\}$

A particular solution role="math" localid="1654061711166" $y_P$of $\left(D-a\right)\left(D-b\right)\mathit{y}\mathbf{=}{\mathit{e}}^{\mathbf{c}\mathbf{x}}{\mathit{P}}_{\mathbf{n}}\left(x\right)$

Where $P_n\left(x\right)$is a polynomial of degree n is

${\mathit{y}}_{\mathbf{p}}\mathbf{=}\{\begin{array}{l}{e}^{cx}{Q}_n\left(x\right)ifc\ne toeitheraorb\\ x{e}^{cx}{Q}_n\left(x\right)ifc=toeitheraorb,a\ne b\\ x^2{e}^{cx}{Q}_n\left(x\right)ifc=a=b\end{array}$

Here $Q_n\left(x\right)$is a polynomial of the degree as $P_n\left(x\right)$with undetermined coefficients to be found to satisfy the given differential equation.