Internal problem ID [9087]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1508.
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 } x^{3}+\left (-\nu ^{2}+1\right ) x y^{\prime }+\left (a \,x^{3}+\nu ^{2}-1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.011 (sec). Leaf size: 81
dsolve(x^3*diff(diff(diff(y(x),x),x),x)+(-nu^2+1)*x*diff(y(x),x)+(a*x^3+nu^2-1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x \hypergeom \left (\left [\right ], \left [\frac {\nu }{3}+1, -\frac {\nu }{3}+1\right ], -\frac {x^{3} a}{27}\right )+c_{2} x^{-\nu +1} \hypergeom \left (\left [\right ], \left [-\frac {\nu }{3}+1, 1-\frac {2 \nu }{3}\right ], -\frac {x^{3} a}{27}\right )+c_{3} x^{\nu +1} \hypergeom \left (\left [\right ], \left [\frac {\nu }{3}+1, \frac {2 \nu }{3}+1\right ], -\frac {x^{3} a}{27}\right ) \]
✓ Solution by Mathematica
Time used: 0.423 (sec). Leaf size: 143
DSolve[(-1 + nu^2 + a*x^3)*y[x] + (1 - nu^2)*x*y'[x] + x^3*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 3^{-\nu -1} x a^{-\nu /3} \left (a^{\frac {\nu +1}{3}} \left (c_3 a^{\nu /3} x^{\nu } \, _0F_2\left (;\frac {\nu }{3}+1,\frac {2 \nu }{3}+1;-\frac {a x^3}{27}\right )+c_1 3^{\nu } \, _0F_2\left (;1-\frac {\nu }{3},\frac {\nu }{3}+1;-\frac {a x^3}{27}\right )\right )+\sqrt [3]{a} c_2 9^{\nu } x^{-\nu } \, _0F_2\left (;1-\frac {2 \nu }{3},1-\frac {\nu }{3};-\frac {a x^3}{27}\right )\right ) \\ \end{align*}