Solution to an Initial Value Problem by Reduction of Order Method

Question

Given: $(1-x^2)y''+2xy'-2y=0$, where $y(0)=3, y'(0)=-4$ and $y_1=x$

My question is, how do I use the initial values for the reduction method? Where do I implement them, and how do I go about it?

Answer 1

Fist you apply the reduction method to find a second linearly independent solution $y_2$ and the general solution $y=C_1\,y_1+C_2\,y_2$. Then use the initial conditions to find $C_1$ and $C_2$.

Variation of parameters is used to find a particular solution of the complete equation once you have the general solution of the homogeneous equation.