ODE
\[ \left (1-a^2 x^2\right ) y''(x)-2 a^2 x y'(x)+2 a^2 y(x)=0 \] ODE Classification
[_Gegenbauer]
Book solution method
TO DO
Mathematica ✓
cpu = 0.167481 (sec), leaf count = 39
\[\left \{\left \{y(x)\to a c_1 x-\frac {1}{2} c_2 (a x \log (1-a x)-a x \log (a x+1)+2)\right \}\right \}\]
Maple ✓
cpu = 0.1 (sec), leaf count = 32
\[\left [y \left (x \right ) = \textit {\_C1} x +\textit {\_C2} \left (\frac {a \ln \left (a x -1\right ) x}{2}-\frac {a \ln \left (a x +1\right ) x}{2}+1\right )\right ]\] Mathematica raw input
DSolve[2*a^2*y[x] - 2*a^2*x*y'[x] + (1 - a^2*x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> a*x*C[1] - (C[2]*(2 + a*x*Log[1 - a*x] - a*x*Log[1 + a*x]))/2}}
Maple raw input
dsolve((-a^2*x^2+1)*diff(diff(y(x),x),x)-2*a^2*x*diff(y(x),x)+2*a^2*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*x+_C2*(1/2*a*ln(a*x-1)*x-1/2*a*ln(a*x+1)*x+1)]