4.44.12 \(x y'''(x)-\left (3-x^2\right ) y''(x)+4 x y'(x)+2 y(x)=0\)

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 = 0.136741 (sec), leaf count = 111

\[\left \{\left \{y(x)\to \frac {1}{240} \left (8 \sqrt {2 \pi } c_2 e^{-\frac {x^2}{2}} x^5 \text {erfi}\left (\frac {x}{\sqrt {2}}\right )+15 c_3 e^{-\frac {x^2}{2}} x^5 \text {Ei}\left (\frac {x^2}{2}\right )-16 c_2 x^4-30 c_3 x^3-16 c_2 x^2+240 c_1 e^{-\frac {x^2}{2}} x^5-60 c_3 x-48 c_2\right )\right \}\right \}\]

Maple ✓
cpu = 0.043 (sec), leaf count = 81

\[ \left \{ y \left ( x \right ) ={{\rm e}^{-{\frac {{x}^{2}}{2}}}}{x}^{5}{\it \_C1}\,{\it Ei} \left ( 1,-{\frac {{x}^{2}}{2}} \right ) -{\frac { \left ( i{\it Erf} \left ( {\frac {i}{2}}\sqrt {2}x \right ) {\it \_C2}\,\sqrt {2}\sqrt {\pi }-30\,{\it \_C3} \right ) {x}^{5}}{30}{{\rm e}^{-{\frac {{x}^{2}}{2}}}}}-{\frac {{x}^{4}{\it \_C2}}{15}}+2\,{x}^{3}{\it \_C1}-{\frac {{x}^{2}{\it \_C2}}{15}}+4\,{\it \_C1}\,x-{\frac {{\it \_C2}}{5}} \right \} \] 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])/E^(x^2/2) - 48*C[2] - 16*x^2*C[2] - 16*x^4*C[2] - 60*x
*C[3] - 30*x^3*C[3] + (8*Sqrt[2*Pi]*x^5*C[2]*Erfi[x/Sqrt[2]])/E^(x^2/2) + (15*x^
5*C[3]*ExpIntegralEi[x^2/2])/E^(x^2/2))/240}}

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),'implicit')

Maple raw output

y(x) = exp(-1/2*x^2)*x^5*_C1*Ei(1,-1/2*x^2)-1/30*(I*erf(1/2*I*2^(1/2)*x)*_C2*2^(
1/2)*Pi^(1/2)-30*_C3)*x^5*exp(-1/2*x^2)-1/15*x^4*_C2+2*x^3*_C1-1/15*x^2*_C2+4*_C
1*x-1/5*_C2