Internal problem ID [6433]
Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 49.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\frac {y^{\prime }}{x}-y x^{3}-x^{4}-\frac {1}{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 28
dsolve(diff(y(x),x$2)-1/x*diff(y(x),x)-x^3*y(x)-x^4-1/x=0,y(x), singsol=all)
\[ y \relax (x ) = x \BesselI \left (\frac {2}{5}, \frac {2 x^{\frac {5}{2}}}{5}\right ) c_{2}+x \BesselK \left (\frac {2}{5}, \frac {2 x^{\frac {5}{2}}}{5}\right ) c_{1}-x \]
✓ Solution by Mathematica
Time used: 0.149 (sec). Leaf size: 316
DSolve[y''[x]-1/x*y'[x]-x^3*y[x]-x^4-1/x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\frac {5 \left (x^{5/2}\right )^{13/5} \text {Gamma}\left (\frac {4}{5}\right ) \text {Gamma}\left (\frac {7}{5}\right ) I_{\frac {2}{5}}\left (\frac {2 x^{5/2}}{5}\right ) \, _1F_2\left (\frac {4}{5};\frac {3}{5},\frac {9}{5};\frac {x^5}{25}\right )}{\text {Gamma}\left (\frac {9}{5}\right )}-\frac {\sqrt [5]{5} \left (x^{5/2}\right )^{7/5} \text {Gamma}\left (\frac {1}{5}\right ) \text {Gamma}\left (\frac {3}{5}\right ) I_{-\frac {2}{5}}\left (\frac {2 x^{5/2}}{5}\right ) \, _1F_2\left (\frac {1}{5};\frac {6}{5},\frac {7}{5};\frac {x^5}{25}\right )}{\text {Gamma}\left (\frac {6}{5}\right )}+\frac {5 \left (x^{5/2}\right )^{3/5} \text {Gamma}\left (-\frac {1}{5}\right ) \text {Gamma}\left (\frac {7}{5}\right ) I_{\frac {2}{5}}\left (\frac {2 x^{5/2}}{5}\right ) \, _1F_2\left (-\frac {1}{5};\frac {3}{5},\frac {4}{5};\frac {x^5}{25}\right )}{\text {Gamma}\left (\frac {4}{5}\right )}+\sqrt [5]{5} x^{5/2} \left (-\frac {x^5 \left (x^{5/2}\right )^{2/5} \text {Gamma}\left (\frac {3}{5}\right ) \text {Gamma}\left (\frac {6}{5}\right ) I_{-\frac {2}{5}}\left (\frac {2 x^{5/2}}{5}\right ) \, _1F_2\left (\frac {6}{5};\frac {7}{5},\frac {11}{5};\frac {x^5}{25}\right )}{\text {Gamma}\left (\frac {11}{5}\right )}+10 \left (c_1 \text {Gamma}\left (\frac {3}{5}\right ) I_{-\frac {2}{5}}\left (\frac {2 x^{5/2}}{5}\right )+(-1)^{2/5} c_2 \text {Gamma}\left (\frac {7}{5}\right ) I_{\frac {2}{5}}\left (\frac {2 x^{5/2}}{5}\right )\right )\right )}{10\ 5^{3/5} x^{3/2}} \\ \end{align*}