Internal problem ID [9058]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1479.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x y^{\prime \prime \prime }+\left (a +b \right ) y^{\prime \prime }-y^{\prime } x -a y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.013 (sec). Leaf size: 92
dsolve(x*diff(diff(diff(y(x),x),x),x)+(a+b)*diff(diff(y(x),x),x)-x*diff(y(x),x)-a*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [\frac {a}{2}\right ], \left [\frac {1}{2}, \frac {a}{2}+\frac {b}{2}\right ], \frac {x^{2}}{4}\right )+c_{2} x \hypergeom \left (\left [\frac {1}{2}+\frac {a}{2}\right ], \left [\frac {3}{2}, \frac {a}{2}+\frac {b}{2}+\frac {1}{2}\right ], \frac {x^{2}}{4}\right )+c_{3} x^{-a -b +2} \hypergeom \left (\left [1-\frac {b}{2}\right ], \left [2-\frac {b}{2}-\frac {a}{2}, -\frac {a}{2}-\frac {b}{2}+\frac {3}{2}\right ], \frac {x^{2}}{4}\right ) \]
✓ Solution by Mathematica
Time used: 0.076 (sec). Leaf size: 153
DSolve[-(a*y[x]) - x*y'[x] + (a + b)*y''[x] + x*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} i c_2 x \, _1F_2\left (\frac {a}{2}+\frac {1}{2};\frac {3}{2},\frac {a}{2}+\frac {b}{2}+\frac {1}{2};\frac {x^2}{4}\right )+c_1 \, _1F_2\left (\frac {a}{2};\frac {1}{2},\frac {a}{2}+\frac {b}{2};\frac {x^2}{4}\right )+c_3 \left (\frac {i}{2}\right )^{-a-b+2} x^{-a-b+2} \, _1F_2\left (1-\frac {b}{2};-\frac {a}{2}-\frac {b}{2}+\frac {3}{2},-\frac {a}{2}-\frac {b}{2}+2;\frac {x^2}{4}\right ) \\ \end{align*}