ODE
\[ \left (1-x^2\right ) y''(x)+x y'(x)+3 y(x)=0 \] ODE Classification
[[_2nd_order, _exact, _linear, _homogeneous]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0560571 (sec), leaf count = 75
\[\left \{\left \{y(x)\to \frac {\left (x^2-1\right )^{3/4} \left (\sqrt {\pi } c_2 \left (1-x^2\right )^{3/4} Q_{\frac {3}{2}}^{\frac {3}{2}}(x)+\sqrt {2} c_1 x \left (2 x^2-3\right )\right )}{\sqrt {\pi } \left (1-x^2\right )^{3/4}}\right \}\right \}\]
Maple ✓
cpu = 0.011 (sec), leaf count = 28
\[ \left \{ y \left ( x \right ) ={\it \_C2}\, \left ( -1+x \right ) ^{{\frac {3}{2}}} \left ( 1+x \right ) ^{{\frac {3}{2}}}+ \left ( 2\,{x}^{3}-3\,x \right ) {\it \_C1} \right \} \] Mathematica raw input
DSolve[3*y[x] + x*y'[x] + (1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> ((-1 + x^2)^(3/4)*(Sqrt[2]*x*(-3 + 2*x^2)*C[1] + Sqrt[Pi]*(1 - x^2)^(3/4)*C[2]*LegendreQ[3/2, 3/2, x]))/(Sqrt[Pi]*(1 - x^2)^(3/4))}}
Maple raw input
dsolve((-x^2+1)*diff(diff(y(x),x),x)+x*diff(y(x),x)+3*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C2*(-1+x)^(3/2)*(1+x)^(3/2)+(2*x^3-3*x)*_C1