Chapter 2.3, Problem 36E

Variation of Parameters. Here is another procedure for solving linear equations that is particularly useful for higher-order linear equations. This method is called variation of parameters. It is based on the idea that just by knowing the form of the solution, we can substitute into the given equation and solve for any unknowns. Here we illustrate the method for first-order equations (see Sections 4.6 and 6.4 for the generalization to higher-order equations)

a. Show that the general solution to

$\frac{dy}{dx}+p\left(x\right)y=Q\left(x\right)$ … $\left(20\right)$

has the form

$y\left(x\right)=C{y}_h\left(x\right)+y_p\left(x\right)$,

Where $y_h\left(\ne 0\right)$ is a solution to equation $\left(20\right)$ when $Q\left(x\right)\equiv 0$, $C$ is a constant, and $y_p\left(x\right)=v\left(x\right)y_h\left(x\right)$ for a suitable function $v\left(x\right)$.[Hint: Show that we can take $y_h={\mu }^{-1}\left(x\right)$ and then use equation $\left(8\right)$.]

We can in fact determine the unknown function $y_h$ by solving a separable equation. Then direct substitution of $v{y}_h$ in the original equation will give a simple equation that can be solved for $v$.

Use this procedure to find the general solution to

$\frac{dy}{dx}+\frac{3}{x}y=x^2$, $x>0$, … $\left(21\right)$

by completing the followings steps:

b. Find a nontrivial solution $y_h$ to the separable equation.

$\frac{dy}{dx}+\frac{3}{x}y=0$, $x>0$, … $\left(22\right)$

c. Assuming $\left(21\right)$ has a solution of the form $y_p\left(x\right)=v\left(x\right)y_h\left(x\right)$, substitute this into equation $\left(21\right)$ and simplify to obtain $v^{\prime }\left(x\right)=x^2/y_h\left(x\right)$.

d. Now integrate to get $v\left(x\right)$.

e. Verify that $y\left(x\right)=C{y}_h\left(x\right)+v\left(x\right)y_h\left(x\right)$ is a general solution to $\left(21\right)$.