Internal problem ID [9103]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1524.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{6} y^{\prime \prime \prime }+y^{\prime \prime } x^{2}-2 y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.813 (sec). Leaf size: 101
dsolve(x^6*diff(diff(diff(y(x),x),x),x)+x^2*diff(diff(y(x),x),x)-2*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{2}+c_{2} \left (\int \frac {{\mathrm e}^{\frac {1}{6 x^{3}}} \left (2 x^{3} \BesselK \left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )+\BesselK \left (\frac {5}{6}, -\frac {1}{6 x^{3}}\right )-\BesselK \left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )\right )}{x^{\frac {11}{2}}}d x \right ) x^{2}+c_{3} \left (\int -\frac {{\mathrm e}^{\frac {1}{6 x^{3}}} \left (-2 x^{3} \BesselI \left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )+\BesselI \left (-\frac {5}{6}, -\frac {1}{6 x^{3}}\right )+\BesselI \left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )\right )}{x^{\frac {11}{2}}}d x \right ) x^{2} \]
✓ Solution by Mathematica
Time used: 0.13 (sec). Leaf size: 74
DSolve[-2*y[x] + x^2*y''[x] + x^6*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\left (-\frac {1}{3}\right )^{2/3} c_2 x \, _2F_2\left (-\frac {2}{3},\frac {1}{3};\frac {2}{3},\frac {4}{3};\frac {1}{3 x^3}\right )+\frac {1}{6} c_3 \, _2F_2\left (-\frac {1}{3},\frac {2}{3};\frac {4}{3},\frac {5}{3};\frac {1}{3 x^3}\right )+c_1 x^2 \\ \end{align*}