Internal problem ID [9081]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1502.
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 }-\left (x^{4}-6 x \right ) y^{\prime \prime }-\left (2 x^{3}-6\right ) y^{\prime }+2 x^{2} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 110
dsolve(x^2*diff(diff(diff(y(x),x),x),x)-(x^4-6*x)*diff(diff(y(x),x),x)-(2*x^3-6)*diff(y(x),x)+2*x^2*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1}}{x^{2}}+\frac {c_{2} \left (\int {\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \left (\BesselI \left (\frac {1}{6}, -\frac {x^{3}}{6}\right ) x^{3}+\BesselI \left (-\frac {5}{6}, -\frac {x^{3}}{6}\right ) x^{3}-2 \BesselI \left (\frac {1}{6}, -\frac {x^{3}}{6}\right )\right )d x \right )}{x^{2}}+\frac {c_{3} \left (\int -{\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \left (-\BesselK \left (\frac {5}{6}, -\frac {x^{3}}{6}\right ) x^{3}+\BesselK \left (\frac {1}{6}, -\frac {x^{3}}{6}\right ) x^{3}-2 \BesselK \left (\frac {1}{6}, -\frac {x^{3}}{6}\right )\right )d x \right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.057 (sec). Leaf size: 75
DSolve[2*x^2*y[x] - (-6 + 2*x^3)*y'[x] - (-6*x + x^4)*y''[x] + x^2*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_2 \, _2F_2\left (-\frac {2}{3},\frac {1}{3};\frac {2}{3},\frac {4}{3};\frac {x^3}{3}\right )}{x}+\frac {1}{2} \sqrt [3]{-\frac {1}{3}} c_3 \, _2F_2\left (-\frac {1}{3},\frac {2}{3};\frac {4}{3},\frac {5}{3};\frac {x^3}{3}\right )+\frac {c_1}{x^2} \\ \end{align*}