ODE
\[ \left (2 x^2+1\right ) y''(x)+3 x y'(x)-3 y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0291386 (sec), leaf count = 55
\[\left \{\left \{y(x)\to c_2 \sqrt [8]{2 x^2+1} Q_{\frac {3}{4}}^{\frac {1}{4}}\left (i \sqrt {2} x\right )+\frac {i 2^{3/4} c_1 x}{\Gamma \left (\frac {3}{4}\right )}\right \}\right \}\]
Maple ✓
cpu = 0.214 (sec), leaf count = 39
\[ \left \{ y \left ( x \right ) =\sqrt [8]{2\,{x}^{2}+1} \left ( {\it LegendreQ} \left ( {\frac {3}{4}},{\frac {1}{4}},i\sqrt {2}x \right ) {\it \_C2}+{\it LegendreP} \left ( {\frac {3}{4}},{\frac {1}{4}},i\sqrt {2}x \right ) {\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[-3*y[x] + 3*x*y'[x] + (1 + 2*x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (I*2^(3/4)*x*C[1])/Gamma[3/4] + (1 + 2*x^2)^(1/8)*C[2]*LegendreQ[3/4, 1/4, I*Sqrt[2]*x]}}
Maple raw input
dsolve((2*x^2+1)*diff(diff(y(x),x),x)+3*x*diff(y(x),x)-3*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (2*x^2+1)^(1/8)*(LegendreQ(3/4,1/4,I*2^(1/2)*x)*_C2+LegendreP(3/4,1/4,I*2^(1/2)*x)*_C1)