Transform the following differential equation into a second order differential equation such that the dependent variable is missing. Solve the corresponding differential equation. $$ x^2 y^{''}-3xy'+4y=x^{1/2} $$
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Perhaps you could find a change of variables $s=f(y,z)$ so that $s'' \propto x^2 y''$ and $s' \propto xy'$? just an idea ... – Matti P. Sep 09 '19 at 10:02
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ok will try to solve it like that – adithya Sep 09 '19 at 10:10
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I would say after substitution
$\ln x = t\\ y(x) = z(\ln x)=z(t)\\ \displaystyle \frac{dy}{dx}=\frac{dz}{dt}\cdot\frac{dt}{dx}=\frac1x\cdot\frac{dz}{dt},\quad \frac{d^2y}{dx^2}=\cdots=\frac1{x^2}\left(\frac{d^2z}{dt^2}-\frac{dz}{dt}\right)$
we get an equation with constant coefficients: $\displaystyle \quad z''-4z'+4z=e^{t/2}$
georg
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