ODE
\[ x^2 \left (-y'(x)\right )+y''(x)+x y(x)=x \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0695336 (sec), leaf count = 52
\[\left \{\left \{y(x)\to -c_2 e^{\frac {x^3}{3}}+\frac {c_2 \sqrt [3]{-x^3} \Gamma \left (\frac {2}{3},-\frac {x^3}{3}\right )}{\sqrt [3]{3}}+c_1 x+1\right \}\right \}\]
Maple ✓
cpu = 0.038 (sec), leaf count = 49
\[ \left \{ y \left ( x \right ) =x{\it \_C2}+{\frac {{\it \_C1}}{{x}^{2}} \left ( \sqrt [3]{3}{{\rm e}^{{\frac {{x}^{3}}{3}}}} \left ( -{x}^{3} \right ) ^{{\frac {2}{3}}}-{x}^{3} \left ( \Gamma \left ( {\frac {2}{3}} \right ) -\Gamma \left ( {\frac {2}{3}},-{\frac {{x}^{3}}{3}} \right ) \right ) \right ) }+1 \right \} \] Mathematica raw input
DSolve[x*y[x] - x^2*y'[x] + y''[x] == x,y[x],x]
Mathematica raw output
{{y[x] -> 1 + x*C[1] - E^(x^3/3)*C[2] + ((-x^3)^(1/3)*C[2]*Gamma[2/3, -x^3/3])/3^(1/3)}}
Maple raw input
dsolve(diff(diff(y(x),x),x)-x^2*diff(y(x),x)+x*y(x) = x, y(x),'implicit')
Maple raw output
y(x) = x*_C2+(3^(1/3)*exp(1/3*x^3)*(-x^3)^(2/3)-x^3*(GAMMA(2/3)-GAMMA(2/3,-1/3*x^3)))/x^2*_C1+1