Variation of Parameters. Here is another procedure for solving linear equations that...

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

has the form

Where yh is a solution to equation (20) when Q (x) =0, C is a constant, and yp (x) = v (x) yh (x) for a suitable function v (x). [Hint: Show that we can take th = µ^-1 (x) and then use equation (8).] We can in fact determine the unknown function yh by solving a separable equation. Then direct substitution of yyh in the original equation will give a simple equation that can be solved for y. Use this procedure to find the general solution to

by completing the following steps:

(b) Find a nontrivial solution yh to the separable equation

(c) Assuming (21) has a solution of the form substitute this into equation

(21), and simplify to obtain

(d) Now integrate to get v (x) .

(e) Verify that y(x) = cyh(x) + v (x)yh (x) is a general solution to (21).