Internal problem ID [9390]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1057.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-x^{2} y^{\prime }-\left (x +1\right )^{2} y=0} \]
✓ Solution by Maple
Time used: 0.344 (sec). Leaf size: 50
dsolve(diff(diff(y(x),x),x)-x^2*diff(y(x),x)-(x+1)^2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {HeunT}\left (0, -3, 2 \,3^{\frac {1}{3}}, \frac {3^{\frac {2}{3}} x}{3}\right ) {\mathrm e}^{-x}+c_{2} \operatorname {HeunT}\left (0, 3, 2 \,3^{\frac {1}{3}}, -\frac {3^{\frac {2}{3}} x}{3}\right ) {\mathrm e}^{\frac {x \left (x^{2}+3\right )}{3}} \]
✓ Solution by Mathematica
Time used: 0.274 (sec). Leaf size: 44
DSolve[-((1 + x)^2*y[x]) - x^2*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{\frac {x^3}{3}+x} \left (c_2 \int _1^xe^{-\frac {1}{3} K[1] \left (K[1]^2+6\right )}dK[1]+c_1\right ) \]