# Anarchy

Calculus Level 5

Given that $$y = f_1(x) = x$$ and $$y = f_2(x)$$ are linearly independent solutions of the differential equation

$(1-x^2)\frac{d^2y}{dx^2} - 2x\frac{dy}{dx} + 2y = 0,$

find $$f_2(x)$$, which satisfies $$f_2 \left( \dfrac{1}{2} \right) = \ln 3 - 4$$.

If $$f_2 \left( \dfrac{3}{4} \right) = \dfrac{a \ln b}{c} - d$$, enter your answer as $$a+b+c+d$$.

Note: $$a, b, c$$ and $$d$$ are positive integers, $$\gcd(a,b) = 1$$, and $$b \neq 1$$ is squarefree.

### This is not an original problem. (Adapted from 'Differential Equations: Linear, Nonlinear, Ordinary, Partial' by A C King, J Billingham, and S R Otto.)

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