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.)


Problem Loading...

Note Loading...

Set Loading...