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 = 0.0334776 (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 = 0.054 (sec), leaf count = 31
\[ \left \{ y \left ( x \right ) ={\frac {{\it \_C2}\,a\ln \left ( ax-1 \right ) x}{2}}-{\frac {{\it \_C2}\,a\ln \left ( ax+1 \right ) x}{2}}+{\it \_C1}\,x+{\it \_C2} \right \} \] 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),'implicit')
Maple raw output
y(x) = 1/2*_C2*a*ln(a*x-1)*x-1/2*_C2*a*ln(a*x+1)*x+_C1*x+_C2