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I've tried so hard and just get horrible, horrible equations. $$y_p = u_1(x)x^{1/2} + u_2(x)x^{1/2}\log x$$

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You might try using reduction of order instead. Knowing $x^{1/2}$ is a solution, let $y = x^{1/2} u(x)$, and you should get $$ x v' + v = \dfrac{2}{x}$$ where $v = u'$.

EDIT: It's unfortunate that some texts restrict reduction of order to the homogeneous case. There's really no reason to do so. Anyway, if you insist on using variation of parameters, the Wronskian of $y_1 = x^{1/2}$ and $y_2 = x^{1/2} \log x$ is $1$, which should make things not so bad.

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We have to use variation of parameters method. But yes, I got x^1/2 and x^1/2logx as my first two solutions when I use reduction of order method, but that was a different question where it was homogeneous. – tom Nov 6 '12 at 23:57

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