How to use the Wronskian to find the in homogeneous solution of an ODE?

Question

Suppose I have an ODE in the form:

$ y'' + p(x)y' + q(x)y = f(x) $

Suppose now that I have obtained the solution to the equation

$ y'' + p(x)y' + q(x)y = 0 $

which is in the form

$ y(x) = c_1 y_1(x) + c_2 y_2(x) $

My textbook suggests that it is possible to use the Wronskian of the solution in order to obtain a solution to the inhomogeneous equation, hence getting the general solution to the equation.

For example, the homogeneous solution to:

$y'' + 6y' + 2y = 1 $

is $ c_1exp(-3+\sqrt 7)x + c_2exp(-3-\sqrt 7)x $

How is then one able to use the Wronskian to obtain the general solution (which I am able to derive ($y_p = 0.5$) by other methods, but not this one).

Answer 1

You can use method of variation of parameters in such cases. The general solution is $$y(x)=p(x)y_1(x)+q(x)y_2(x),$$ where $p(x)=-\int \dfrac{y_2(x)f(x)}{w(y_1,y_2)} dx+c_1$ and $q(x)=\int \dfrac{y_1(x)f(x)}{w(y_1,y_2)} dx+c_2$ and $w(y_1,y_2)$ be the Wronskian of $y_1(x)$ and $y_2(x)$.