Internal problem ID [9076]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1497.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime \prime }-3 \left (p +q \right ) x y^{\prime \prime }+3 p \left (3 q +1\right ) y^{\prime }-x^{2} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 77
dsolve(x^2*diff(diff(diff(y(x),x),x),x)-3*(p+q)*x*diff(diff(y(x),x),x)+3*p*(3*q+1)*diff(y(x),x)-x^2*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [\right ], \left [-q +\frac {1}{3}, -p +\frac {2}{3}\right ], \frac {x^{3}}{27}\right )+c_{2} x^{3 p +1} \hypergeom \left (\left [\right ], \left [p +\frac {4}{3}, \frac {2}{3}-q +p \right ], \frac {x^{3}}{27}\right )+c_{3} x^{3 q +2} \hypergeom \left (\left [\right ], \left [q +\frac {5}{3}, \frac {4}{3}+q -p \right ], \frac {x^{3}}{27}\right ) \]
✓ Solution by Mathematica
Time used: 0.259 (sec). Leaf size: 127
DSolve[-(x^2*y[x]) + 3*p*(1 + 3*q)*y'[x] - 3*(p + q)*x*y''[x] + x^2*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \, _0F_2\left (;\frac {2}{3}-p,\frac {1}{3}-q;\frac {x^3}{27}\right )+c_2 (-1)^{p+\frac {1}{3}} 3^{-3 p-1} x^{3 p+1} \, _0F_2\left (;p+\frac {4}{3},p-q+\frac {2}{3};\frac {x^3}{27}\right )+c_3 (-1)^{q+\frac {2}{3}} 3^{-3 q-2} x^{3 q+2} \, _0F_2\left (;q+\frac {5}{3},-p+q+\frac {4}{3};\frac {x^3}{27}\right ) \\ \end{align*}