ODE
\[ y'''(x)+x^2 \left (-y''(x)\right )+2 x y'(x)-2 y(x)=0 \] ODE Classification
[[_3rd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.217558 (sec), leaf count = 79
\[\left \{\left \{y(x)\to \frac {1}{18} \left (-\sqrt [3]{3} c_3 \left (-x^3\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {x^3}{3}\right )+3^{2/3} c_3 \sqrt [3]{-x^3} \Gamma \left (-\frac {1}{3},-\frac {x^3}{3}\right )+9 x (c_2 x+2 c_1)\right )\right \}\right \}\]
Maple ✓
cpu = 0.258 (sec), leaf count = 109
\[\left [y \left (x \right ) = \textit {\_C1} x +x^{2} \textit {\_C2} +\frac {\textit {\_C3} \left (-2 \,3^{\frac {5}{6}} x^{3} \pi \left (-x^{3}\right )^{\frac {1}{3}}+6 \,3^{\frac {2}{3}} x^{3} \Gamma \left (\frac {2}{3}\right )^{2}-6 \,3^{\frac {2}{3}} x^{3} \Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right ) \Gamma \left (\frac {2}{3}\right )+3 \,3^{\frac {1}{3}} x^{3} \Gamma \left (\frac {1}{3}, -\frac {x^{3}}{3}\right ) \left (-x^{3}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}\right )-9 \,{\mathrm e}^{\frac {x^{3}}{3}} \left (-x^{3}\right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right )\right )}{\left (-x^{3}\right )^{\frac {2}{3}}}\right ]\] Mathematica raw input
DSolve[-2*y[x] + 2*x*y'[x] - x^2*y''[x] + y'''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (9*x*(2*C[1] + x*C[2]) - 3^(1/3)*(-x^3)^(2/3)*C[3]*Gamma[-2/3, -1/3*x^3] + 3^(2/3)*(-x^3)^(1/3)*C[3]*Gamma[-1/3, -1/3*x^3])/18}}
Maple raw input
dsolve(diff(diff(diff(y(x),x),x),x)-x^2*diff(diff(y(x),x),x)+2*x*diff(y(x),x)-2*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*x+x^2*_C2+_C3*(-2*3^(5/6)*x^3*Pi*(-x^3)^(1/3)+6*3^(2/3)*x^3*GAMMA(2/3)^2-6*3^(2/3)*x^3*GAMMA(2/3,-1/3*x^3)*GAMMA(2/3)+3*3^(1/3)*x^3*GAMMA(1/3,-1/3*x^3)*(-x^3)^(1/3)*GAMMA(2/3)-9*exp(1/3*x^3)*(-x^3)^(2/3)*GAMMA(2/3))/(-x^3)^(2/3)]