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I come across a second order ODE recently of the form

$$ Ax^2 f''(x) + Bx f'(x) + C f(x) = 0,$$

where $A,B,C$ are constants. The method introduced in most textbooks only covers ODEs with constant coefficients. Are there any exact solutions that I could use, rather than those series solutions?

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This particular case can be easily solved with the change of variables $x = e^t$ and introducing $z(t)=f(e^t)$. Then you can write the differential equation on $z$, which yields a linear ODE with constant coeffients. Then you come back to $f$ and $x$ by the inverse change of variables.

Another approach would be to say from the very beginning that we will seek the solutions in the form $f(x) = x^a$ with $a$ - constant, and then obtain a quadratic equation on $a$, which leaves you with two cases: distinct roots give you two different solutions $x^{a_1}$ and $x^{a_2}$; equal roots lead to solutions $x^{a}$ and $x^a\ln x$.

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