Internal problem ID [9030]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1451.
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 }-a \,x^{b} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 114
dsolve(diff(diff(diff(y(x),x),x),x)-a*x^b*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [\right ], \left [\frac {b +1}{b +3}, \frac {b +2}{b +3}\right ], \frac {a \,x^{b +3}}{\left (b +3\right )^{3}}\right )+c_{2} x \hypergeom \left (\left [\right ], \left [\frac {b +2}{b +3}, \frac {b +4}{b +3}\right ], \frac {a \,x^{b +3}}{\left (b +3\right )^{3}}\right )+c_{3} x^{2} \hypergeom \left (\left [\right ], \left [\frac {b +5}{b +3}, \frac {b +4}{b +3}\right ], \frac {a \,x^{b +3}}{\left (b +3\right )^{3}}\right ) \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 164
DSolve[-(a*x^b*y[x]) + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to (-1)^{\frac {1}{b+3}} (b+3)^{-\frac {6}{b+3}} x a^{\frac {1}{b+3}} \left ((-1)^{\frac {1}{b+3}} c_3 x a^{\frac {1}{b+3}} \, _0F_2\left (;1+\frac {1}{b+3},1+\frac {2}{b+3};\frac {a x^{b+3}}{(b+3)^3}\right )+(b+3)^{\frac {3}{b+3}} c_2 \, _0F_2\left (;1-\frac {1}{b+3},1+\frac {1}{b+3};\frac {a x^{b+3}}{(b+3)^3}\right )\right )+c_1 \, _0F_2\left (;1-\frac {2}{b+3},1-\frac {1}{b+3};\frac {a x^{b+3}}{(b+3)^3}\right ) \\ \end{align*}