ODE
\[ x y'''(x)-\left (3-x^2\right ) y''(x)+4 x y'(x)+2 y(x)=0 \] ODE Classification
[[_3rd_order, _fully, _exact, _linear]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.284464 (sec), leaf count = 103
\[\left \{\left \{y(x)\to \frac {1}{240} e^{-\frac {x^2}{2}} \left (8 c_2 \left (\sqrt {2 \pi } x^5 \text {erfi}\left (\frac {x}{\sqrt {2}}\right )-2 e^{\frac {x^2}{2}} \left (x^4+x^2+3\right )\right )+15 c_3 x \left (x^4 \text {Ei}\left (\frac {x^2}{2}\right )-2 e^{\frac {x^2}{2}} \left (x^2+2\right )\right )+240 c_1 x^5\right )\right \}\right \}\]
Maple ✓
cpu = 0.183 (sec), leaf count = 83
\[\left [y \left (x \right ) = x \left ({\mathrm e}^{-\frac {x^{2}}{2}} x^{4} \expIntegral \left (1, -\frac {x^{2}}{2}\right )+2 x^{2}+4\right ) \textit {\_C1} +\left (-\frac {1}{5}-\frac {x^{2}}{15}-\frac {x^{4}}{15}-\frac {i {\mathrm e}^{-\frac {x^{2}}{2}} x^{5} \sqrt {\pi }\, \sqrt {2}\, \erf \left (\frac {i \sqrt {2}\, x}{2}\right )}{30}\right ) \textit {\_C2} +{\mathrm e}^{-\frac {x^{2}}{2}} x^{5} \textit {\_C3}\right ]\] Mathematica raw input
DSolve[2*y[x] + 4*x*y'[x] - (3 - x^2)*y''[x] + x*y'''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (240*x^5*C[1] + 8*C[2]*(-2*E^(x^2/2)*(3 + x^2 + x^4) + Sqrt[2*Pi]*x^5*Erfi[x/Sqrt[2]]) + 15*x*C[3]*(-2*E^(x^2/2)*(2 + x^2) + x^4*ExpIntegralEi[x^2/2]))/(240*E^(x^2/2))}}
Maple raw input
dsolve(x*diff(diff(diff(y(x),x),x),x)-(-x^2+3)*diff(diff(y(x),x),x)+4*x*diff(y(x),x)+2*y(x) = 0, y(x))
Maple raw output
[y(x) = x*(exp(-1/2*x^2)*x^4*Ei(1,-1/2*x^2)+2*x^2+4)*_C1+(-1/5-1/15*x^2-1/15*x^4-1/30*I*exp(-1/2*x^2)*x^5*Pi^(1/2)*2^(1/2)*erf(1/2*I*2^(1/2)*x))*_C2+exp(-1/2*x^2)*x^5*_C3]