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I am looking through some solved exercises. One of them is the following:

Solve the equation: x^2 y'' + (x^2 - 3x)y' + (3-x)y = x^4

knowing that y=x is a solution of the homogeneous equation.

The professor then solves it by doing the following substitution: y=xz.

Then he calculates y' and y'', substitutes, etc.

What I do not understand is, how do you know, from the fact that y=x is a solution to the homogeneous equation, that you have to do that substitution y=xz?

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# Simple differential equation substitution

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