Internal problem ID [9111]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1532.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime }+y^{\prime } x +n y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 58
dsolve(diff(diff(diff(y(x),x),x),x)+x*diff(y(x),x)+n*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [\frac {n}{3}\right ], \left [\frac {1}{3}, \frac {2}{3}\right ], -\frac {x^{3}}{9}\right )+c_{2} x \hypergeom \left (\left [\frac {1}{3}+\frac {n}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], -\frac {x^{3}}{9}\right )+c_{3} x^{2} \hypergeom \left (\left [\frac {2}{3}+\frac {n}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], -\frac {x^{3}}{9}\right ) \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 103
DSolve[n*y[x] + x*y'[x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_2 x \, _1F_2\left (\frac {n}{3}+\frac {1}{3};\frac {2}{3},\frac {4}{3};-\frac {x^3}{9}\right )}{3^{2/3}}+c_1 \, _1F_2\left (\frac {n}{3};\frac {1}{3},\frac {2}{3};-\frac {x^3}{9}\right )+\frac {c_3 x^2 \, _1F_2\left (\frac {n}{3}+\frac {2}{3};\frac {4}{3},\frac {5}{3};-\frac {x^3}{9}\right )}{3 \sqrt [3]{3}} \\ \end{align*}