Internal problem ID [6428]
Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 44.
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^{2}-x y-x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 44
dsolve(diff(y(x),x$2)-x^2*diff(y(x),x)-x*y(x)-x^2=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \BesselI \left (\frac {1}{6}, \frac {x^{3}}{6}\right ) c_{2}+{\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \BesselK \left (\frac {1}{6}, \frac {x^{3}}{6}\right ) c_{1}-\frac {x}{2} \]
✓ Solution by Mathematica
Time used: 0.245 (sec). Leaf size: 224
DSolve[y''[x]-x^2*y'[x]-x*y[x]-x^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {e^{\frac {x^3}{6}} \left (12 \left (x^3\right )^{5/6} \text {Gamma}\left (\frac {1}{6}\right ) \text {Gamma}\left (\frac {7}{6}\right ) I_{\frac {1}{6}}\left (\frac {x^3}{6}\right ) \, _1F_1\left (-\frac {2}{3};-\frac {1}{3};-\frac {x^3}{3}\right )+\sqrt [3]{2} 3^{2/3} \sqrt [6]{x^3} x^6 \text {Gamma}\left (\frac {1}{6}\right ) \text {Gamma}\left (\frac {5}{6}\right ) I_{-\frac {1}{6}}\left (\frac {x^3}{6}\right ) \, _1F_1\left (\frac {2}{3};\frac {7}{3};-\frac {x^3}{3}\right )-4 \text {Gamma}\left (\frac {7}{6}\right ) \left (6 \sqrt [3]{2} 3^{2/3} c_1 x^{5/2} \text {Gamma}\left (\frac {5}{6}\right ) I_{-\frac {1}{6}}\left (\frac {x^3}{6}\right )+\text {Gamma}\left (\frac {1}{6}\right ) \left (3 \left (x^3\right )^{5/6}+2 \sqrt [3]{-1} 3^{2/3} c_2 x^{5/2}\right ) I_{\frac {1}{6}}\left (\frac {x^3}{6}\right )\right )\right )}{24\ 2^{2/3} 3^{5/6} x^2 \text {Gamma}\left (\frac {7}{6}\right )} \\ \end{align*}