Internal problem ID [9072]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1493.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime \prime }+4 y^{\prime \prime } x +\left (x^{2}+2\right ) y^{\prime }+3 y x -f \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 1849
dsolve(x^2*diff(diff(diff(y(x),x),x),x)+4*x*diff(diff(y(x),x),x)+(x^2+2)*diff(y(x),x)+3*x*y(x)-f(x)=0,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 0.69 (sec). Leaf size: 310
DSolve[-f[x] + 3*x*y[x] + (2 + x^2)*y'[x] + 4*x*y''[x] + x^2*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {(2-\pi x \pmb {H}_0(x)) \int _1^x\frac {9 f(K[3]) K[3]}{16 \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[3]^2\right ) K[3]^4-9 \left (K[3]^2+1\right ) (\pi K[3] \pmb {H}_0(K[3])-2)}dK[3]+x J_0(x) \left (\int _1^x\frac {9 \pi f(K[1]) \left (Y_1(K[1]) K[1] (\pi K[1] \pmb {H}_0(K[1])-2)+Y_0(K[1]) \left (-\pi \pmb {H}_1(K[1]) K[1]^2+2 K[1]^2+2\right )\right )}{2 \left (9 \left (K[1]^2+1\right ) (\pi K[1] \pmb {H}_0(K[1])-2)-16 \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[1]^2\right ) K[1]^4\right )}dK[1]+c_1\right )+2 x Y_0(x) \left (\int _1^x\frac {9 \pi f(K[2]) \left (J_1(K[2]) K[2] (\pi K[2] \pmb {H}_0(K[2])-2)+J_0(K[2]) \left (-\pi \pmb {H}_1(K[2]) K[2]^2+2 K[2]^2+2\right )\right )}{64 \, _1F_2\left (3;\frac {5}{2},\frac {5}{2};-\frac {1}{4} K[2]^2\right ) K[2]^4-36 \left (K[2]^2+1\right ) (\pi K[2] \pmb {H}_0(K[2])-2)}dK[2]+c_2\right )+c_3 (2-\pi x \pmb {H}_0(x))}{x} \\ \end{align*}