Concept:
Given a 2nd order differential equation of the form:
$$y''+py'+qy=0 $$
...and where one solution to the differential equation ,(y1), is known,
...there exists a function (v) where:
$$v' = e^{-\int{}{}(\frac{2y'_1}{y_1}+p)dx} $$
...such that a second solution to the differential equation (y2) is:
$$y_2 = vy_1 $$
...and a general solution to the differential equation is represented by:
$$y_g = C_1y_1+ C_2y_2 $$
(Ex 1): Use D'Alembert's method to find a second solution to the following:
$$x^2y''+3xy+y=0 $$
where one solution (y1) is known:
$$y_1 = \frac{1}{x} $$
First put the the equation in the correct form by dividing through by x^2:
$$y''+\frac{3}{x}y'+\frac{y}{x^2}=0 $$
Now use the above described method to find a function v'
$$v' = e^{-\int{}{}(\frac{2y'_1}{y_1}+p)dx} $$
...where (for our problem):
$$y_1=\frac{1}{x} \;,\; y'_1=\frac{-1}{x^2} \;,\; p=\frac{3}{x}$$
$$v' = e^{-\int{}{}(\frac{-2x^{-2}}{x^{-1}}+\frac{3}{x})dx} $$
$$v' = e^{-\int{\frac{-2}{x} + \frac{3}{x}}{}dx} $$
$$v' = e^{-\int{}{}\frac{1}{x}dx} $$
$$v' = e^{-ln(x)} $$
$$v' = e^{ln(x^{-1})} $$
$$v' = e^{ln(\frac{1}{x})} $$
$$v' = \frac{1}{x} $$
Now integrate to find (v)
$$v = \int{}{}\frac{1}{x} dx $$
$$v = ln(x) $$
...and recally our definition for a second solution to our differential equation:
$$y_2 = vy_1 $$
$$y_2 = ln(x)\frac{1}{x}=\frac{1}{x}ln(x) $$
...and the general solution to our differential equation is represented by:
$$y_g = C_1y_1+C_2y_2 $$
$$y_G = \frac{C_1}{x} + \frac{C_2}{x}ln(x) $$