Question

Question: If y₁ and y₂ are linearly independent solutions of the associated homogeneous DE for ${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{+}\mathit{P}\mathbf{\left(}\mathit{x}\mathbf{\right)}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathit{Q}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathit{y}\mathbf{=}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}$, show in the case of a nonhomogeneous linear second-order DEthat (9) reduces to the form of variation of parameters discussed in Section 4.6.

Short answer

It is proved that the integral form of variation of parameters is $u_1=\int \frac{-y_{2}f\left(x\right)}{W}dx$ and$u_2=\int \frac{y_{1}f\left(x\right)}{W}dx$.

Step-by-step solution

Determine the Nonhomogeneous Linear System

Nonhomogeneous differential equations are the same as homogeneous differential equations, except that they might contain terms on the right side that involve only (and constants) as$y^{\text{'}\text{'}}+P\left(x\right)y^{\text{'}}+Q\left(x\right)y=f\left(x\right)$.

Determine the Nonhomogeneous Linear Second Order

Let us have the differential equation in the form as:

$$\begin{gathered} y^{\text{'}}=v \\ {v}^{\text{'}}=-Qy-Pv+f \end{gathered}$$

It can be written in matrix form as follows:

Now, using the particular solution as:

$X_p=\varphi \left(x\right)\int {\varphi }^{-1}\left(x\right)F\left(x\right)dx$

Therefore, we have a particular solution given as:

Now comparing both expressions for X_p, we have:
$$\begin{gathered} u_1=\int \frac{-y_{2}f\left(x\right)}{W}dx \\ {u}_2=\int \frac{y_{1}f\left(x\right)}{W}dx \end{gathered}$$
These are the integral forms of variation of parameters.
Hence, it is proved that the integral form of variation of parameters is $u_1=\int \frac{-y_{2}f\left(x\right)}{W}dx$and $u_2=\int \frac{y_{1}f\left(x\right)}{W}dx$