Internal problem ID [6435]
Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 51.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-y^{\prime } x^{3}-x^{2} y-x^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.562 (sec). Leaf size: 30
dsolve(diff(y(x),x$2)-x^3*diff(y(x),x)-x^2*y(x)-x^3=0,y(x), singsol=all)
\[ y \relax (x ) = \KummerM \left (\frac {1}{2}, \frac {5}{4}, \frac {x^{4}}{4}\right ) x c_{2}+\KummerU \left (\frac {1}{2}, \frac {5}{4}, \frac {x^{4}}{4}\right ) x c_{1}-\frac {x}{2} \]
✓ Solution by Mathematica
Time used: 0.802 (sec). Leaf size: 274
DSolve[y''[x]-x^3*y'[x]-x^2*y[x]-x^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \, _1F_1\left (\frac {1}{4};\frac {3}{4};\frac {x^4}{4}\right ) \left (\int _1^x\frac {1}{\, _1F_1\left (\frac {1}{4};\frac {3}{4};\frac {K[1]^4}{4}\right ) \left (\frac {3 \, _1F_1\left (\frac {1}{2};\frac {9}{4};\frac {K[1]^4}{4}\right )}{5 \, _1F_1\left (\frac {1}{2};\frac {5}{4};\frac {K[1]^4}{4}\right )}-\frac {1}{K[1]^4}\right )-\frac {2}{3} \, _1F_1\left (\frac {1}{4};\frac {7}{4};\frac {K[1]^4}{4}\right )}dK[1]+c_1\right )+\frac {1}{2} x \, _1F_1\left (\frac {1}{2};\frac {5}{4};\frac {x^4}{4}\right ) \left (\sqrt [4]{-1} \sqrt {2} \int _1^x\frac {(15-15 i) \, _1F_1\left (\frac {1}{4};\frac {3}{4};\frac {K[2]^4}{4}\right ) K[2]^3}{3 \, _1F_1\left (\frac {1}{4};\frac {3}{4};\frac {K[2]^4}{4}\right ) \left (2 \, _1F_1\left (\frac {3}{2};\frac {9}{4};\frac {K[2]^4}{4}\right ) K[2]^4+5 \, _1F_1\left (\frac {1}{2};\frac {5}{4};\frac {K[2]^4}{4}\right )\right )-5 \, _1F_1\left (\frac {1}{2};\frac {5}{4};\frac {K[2]^4}{4}\right ) \, _1F_1\left (\frac {5}{4};\frac {7}{4};\frac {K[2]^4}{4}\right ) K[2]^4}dK[2]+(1+i) c_2\right ) \\ \end{align*}