Internal problem ID [9112]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1533.
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: 0.005 (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: 106
DSolve[-(n*y[x]) - x*y'[x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{9} \left (3 \sqrt [3]{-3} c_2 x \, _1F_2\left (\frac {n}{3}+\frac {1}{3};\frac {2}{3},\frac {4}{3};\frac {x^3}{9}\right )+9 c_1 \, _1F_2\left (\frac {n}{3};\frac {1}{3},\frac {2}{3};\frac {x^3}{9}\right )+(-3)^{2/3} c_3 x^2 \, _1F_2\left (\frac {n}{3}+\frac {2}{3};\frac {4}{3},\frac {5}{3};\frac {x^3}{9}\right )\right ) \\ \end{align*}