# Differential Equation : Basis

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Verify that the given functions form a basis for the space of solutions of the given differential equation:

x^2 y'' - 2xy' + 2y = 0, f_1(x) = x, f_2(x) = x^2, x > 0

https://brainmass.com/math/calculus-and-analysis/differential-equation-basis-16074

#### Solution Preview

Proof. First, we will verify that f_1(x)=x, f_2(x)=x^2 satisfy the differential equation x^2y''-2xy'+2y=0

(1) y=f_1(x)=x

y'=1, y''=0

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#### Solution Summary

It is verified that functions form a basis for the space of solutions of a differential equation.

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