ODE
\[ -\left (2 x^2+1\right ) y'(x)+2 x y''(x)-x y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0273022 (sec), leaf count = 55
\[\left \{\left \{y(x)\to \frac {1}{2} e^{\frac {x^2}{2}} \left (2 c_1-\frac {2^{3/4} c_2 \sqrt [4]{x^2} \Gamma \left (\frac {3}{4},\frac {x^2}{2}\right )}{\sqrt {x}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.217 (sec), leaf count = 29
\[ \left \{ y \left ( x \right ) ={x}^{{\frac {3}{2}}} \left ( {{\sl M}\left (1,\,{\frac {7}{4}},\,{\frac {{x}^{2}}{2}}\right )}{\it \_C1}+{{\sl U}\left (1,\,{\frac {7}{4}},\,{\frac {{x}^{2}}{2}}\right )}{\it \_C2} \right ) \right \} \] Mathematica raw input
DSolve[-(x*y[x]) - (1 + 2*x^2)*y'[x] + 2*x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (E^(x^2/2)*(2*C[1] - (2^(3/4)*(x^2)^(1/4)*C[2]*Gamma[3/4, x^2/2])/Sqrt[x]))/2}}
Maple raw input
dsolve(2*x*diff(diff(y(x),x),x)-(2*x^2+1)*diff(y(x),x)-x*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = x^(3/2)*(KummerM(1,7/4,1/2*x^2)*_C1+KummerU(1,7/4,1/2*x^2)*_C2)