Find y as a function of x if:
(x^2)(y'') + 19xy' +81y = x^2
y(1) = 9 y'(1) = -3
Hint: First assume that at least one solution to the corresponding homogeneous equation is of the form . You may have to use some other method to find the second solution to make a fundamental set of solutions. Then use one of the two methods to find a particular solution.
This is a Cauchy-Euler equation. To solve that we let x=exp(z) and plug this into the equation. In general with this change of variable the equation:
x^2y''+ axy'+ by= f(x) becomes:
y''+ (a-1)y'+ by= f(exp(z)) ...
A homogenous 2nd order ODE is solved.