# Solve a homogenous 2nd order ODE.

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.

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#### Solution Preview

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)) ...

#### Solution Summary

A homogenous 2nd order ODE is solved. The expert solves a homogenous 2nd order ODE. A Cauchy-Euler equation is examined.

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