Internal problem ID [9849]
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]]
\[ \boxed {x^{6} y^{\prime \prime \prime }+x^{2} y^{\prime \prime }-2 y=0} \]
✓ Solution by Maple
Time used: 0.14 (sec). Leaf size: 98
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)
\[ y \left (x \right ) = x^{2} \left (c_{1} +\left (\int \frac {{\mathrm e}^{\frac {1}{6 x^{3}}} \left (2 x^{3} \operatorname {BesselI}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )-\operatorname {BesselI}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )-\operatorname {BesselI}\left (-\frac {5}{6}, -\frac {1}{6 x^{3}}\right )\right )}{x^{\frac {11}{2}}}d x \right ) c_{2} +\left (\int \frac {{\mathrm e}^{\frac {1}{6 x^{3}}} \left (2 x^{3} \operatorname {BesselK}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )-\operatorname {BesselK}\left (\frac {1}{6}, -\frac {1}{6 x^{3}}\right )+\operatorname {BesselK}\left (\frac {5}{6}, -\frac {1}{6 x^{3}}\right )\right )}{x^{\frac {11}{2}}}d x \right ) c_{3} \right ) \]
✓ Solution by Mathematica
Time used: 0.19 (sec). Leaf size: 96
DSolve[-2*y[x] + x^2*y''[x] + x^6*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {\left (-\frac {1}{3}\right )^{2/3} c_2 x \operatorname {Gamma}\left (\frac {1}{3}\right ) \, _2F_2\left (-\frac {2}{3},\frac {1}{3};\frac {2}{3},\frac {4}{3};\frac {1}{3 x^3}\right )}{3 \operatorname {Gamma}\left (\frac {4}{3}\right )}+\frac {c_3 \operatorname {Gamma}\left (\frac {2}{3}\right ) \, _2F_2\left (-\frac {1}{3},\frac {2}{3};\frac {4}{3},\frac {5}{3};\frac {1}{3 x^3}\right )}{9 \operatorname {Gamma}\left (\frac {5}{3}\right )}+c_1 x^2 \]