Internal problem ID [9060]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1481.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _fully, _exact, _linear]]
Solve \begin {gather*} \boxed {x y^{\prime \prime \prime }+\left (x^{2}-3\right ) y^{\prime \prime }+4 y^{\prime } x +2 y-f \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 44
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)-f(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \left (c_{3}+\int \frac {\left (2 c_{1} x +c_{2}-\left (\int \left (\int -f \relax (x )d x \right )d x \right )\right ) {\mathrm e}^{\frac {x^{2}}{2}}}{x^{6}}d x \right ) {\mathrm e}^{-\frac {x^{2}}{2}} x^{5} \]
✓ Solution by Mathematica
Time used: 0.276 (sec). Leaf size: 252
DSolve[-f[x] + 2*y[x] + 4*x*y'[x] + (-3 + x^2)*y''[x] + x*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{240} \left (e^{-\frac {x^2}{2}} x^5 \left (240 \int _1^x\frac {1}{240} f(K[1]) \left (\frac {2 e^{\frac {K[1]^2}{2}} \left (8 \sqrt {2} F\left (\frac {K[1]}{\sqrt {2}}\right ) K[1]^5-8 K[1]^4+7 K[1]^2+6\right )}{K[1]^4}-15 \text {Ei}\left (\frac {K[1]^2}{2}\right )\right )dK[1]+8 \sqrt {2 \pi } \text {Erfi}\left (\frac {x}{\sqrt {2}}\right ) \int _1^x-f(K[2]) K[2]dK[2]+15 \text {Ei}\left (\frac {x^2}{2}\right ) \int _1^xf(K[3])dK[3]+8 \sqrt {2 \pi } c_2 \text {Erfi}\left (\frac {x}{\sqrt {2}}\right )+15 c_3 \text {Ei}\left (\frac {x^2}{2}\right )+240 c_1\right )-30 \left (x^2+2\right ) x \int _1^xf(K[3])dK[3]-16 \left (x^4+x^2+3\right ) \int _1^x-f(K[2]) K[2]dK[2]-30 c_3 \left (x^2+2\right ) x-16 c_2 \left (x^4+x^2+3\right )\right ) \\ \end{align*}