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 = 0.169123 (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 = 0.198 (sec), leaf count = 47
\[\left [y \left (x \right ) = \textit {\_C1} \left (2 x^{2}+1\right )^{\frac {1}{8}} \LegendreP \left (\frac {3}{4}, \frac {1}{4}, i \sqrt {2}\, x \right )+\textit {\_C2} \left (2 x^{2}+1\right )^{\frac {1}{8}} \LegendreQ \left (\frac {3}{4}, \frac {1}{4}, i \sqrt {2}\, x \right )\right ]\] 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))
Maple raw output
[y(x) = _C1*(2*x^2+1)^(1/8)*LegendreP(3/4,1/4,I*2^(1/2)*x)+_C2*(2*x^2+1)^(1/8)*LegendreQ(3/4,1/4,I*2^(1/2)*x)]