4.33.8 $(2x^2+1)y^{″}(x)+3x{y}^{\prime }(x)-3y(x)=0$

ODE
$$(2x^2+1)y^{″}(x)+3x{y}^{\prime }(x)-3y(x)=0$$ ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0291386 (sec), leaf count = 55

$$\left\{\{y(x)\to c_2\sqrt[8]{2x^2+1}{Q}_{\frac{3}{4}}^{\frac{1}{4}}\left(i\sqrt{2}x\right)+\frac{i{2}^{3/4}{c}_{1}x}{\mathrm{\Gamma }\left(\frac{3}{4}\right)}\}\right\}$$

Maple ✓
cpu = 0.214 (sec), leaf count = 39

$$\{y\left(x\right)=\sqrt[8]{2x^2+1}(\mathit{L}\mathit{e}\mathit{g}\mathit{e}\mathit{n}\mathit{d}\mathit{r}\mathit{e}\mathit{Q}(\frac{3}{4},\frac{1}{4},i\sqrt{2}x)\mathit{\_}\mathit{C}\mathit{2}+\mathit{L}\mathit{e}\mathit{g}\mathit{e}\mathit{n}\mathit{d}\mathit{r}\mathit{e}\mathit{P}(\frac{3}{4},\frac{1}{4},i\sqrt{2}x)\mathit{\_}\mathit{C}\mathit{1})\}$$ 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)