Internal problem ID [9034]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1455.
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 \prime } x^{2}+\left (a +b -1\right ) x y^{\prime }-b y a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 71
dsolve(diff(diff(diff(y(x),x),x),x)-x^2*diff(diff(y(x),x),x)+(a+b-1)*x*diff(y(x),x)-b*y(x)*a=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [-\frac {a}{3}, -\frac {b}{3}\right ], \left [\frac {1}{3}, \frac {2}{3}\right ], \frac {x^{3}}{3}\right )+c_{2} \hypergeom \left (\left [\frac {1}{3}-\frac {a}{3}, \frac {1}{3}-\frac {b}{3}\right ], \left [\frac {2}{3}, \frac {4}{3}\right ], \frac {x^{3}}{3}\right ) x +c_{3} \hypergeom \left (\left [-\frac {a}{3}+\frac {2}{3}, -\frac {b}{3}+\frac {2}{3}\right ], \left [\frac {4}{3}, \frac {5}{3}\right ], \frac {x^{3}}{3}\right ) x^{2} \]
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 127
DSolve[-(a*b*y[x]) + (-1 + a + b)*x*y'[x] - x^2*y''[x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt [3]{-\frac {1}{3}} c_2 x \, _2F_2\left (\frac {1}{3}-\frac {a}{3},\frac {1}{3}-\frac {b}{3};\frac {2}{3},\frac {4}{3};\frac {x^3}{3}\right )+c_1 \, _2F_2\left (-\frac {a}{3},-\frac {b}{3};\frac {1}{3},\frac {2}{3};\frac {x^3}{3}\right )+\left (-\frac {1}{3}\right )^{2/3} c_3 x^2 \, _2F_2\left (\frac {2}{3}-\frac {a}{3},\frac {2}{3}-\frac {b}{3};\frac {4}{3},\frac {5}{3};\frac {x^3}{3}\right ) \\ \end{align*}