Internal problem ID [9055]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1478.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime \prime }+3 y^{\prime \prime }-a \,x^{2} y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 48
dsolve(x*diff(diff(diff(y(x),x),x),x)+3*diff(diff(y(x),x),x)-a*x^2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {3}{4}, \frac {5}{4}\right ], \frac {a \,x^{4}}{64}\right )+\frac {c_{2} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {1}{2}, \frac {3}{4}\right ], \frac {a \,x^{4}}{64}\right )}{x}+c_{3} x \operatorname {hypergeom}\left (\left [\right ], \left [\frac {5}{4}, \frac {3}{2}\right ], \frac {a \,x^{4}}{64}\right ) \]
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 90
DSolve[-(a*x^2*y[x]) + 3*y''[x] + x*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {(2-2 i) c_1 \, _0F_2\left (;\frac {1}{2},\frac {3}{4};\frac {a x^4}{64}\right )}{\sqrt [4]{a} x}+c_2 \, _0F_2\left (;\frac {3}{4},\frac {5}{4};\frac {a x^4}{64}\right )+\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt [4]{a} c_3 x \, _0F_2\left (;\frac {5}{4},\frac {3}{2};\frac {a x^4}{64}\right ) \\ \end{align*}