60.3.56 problem 1057

Internal problem ID [11066]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1057
Date solved : Monday, January 27, 2025 at 10:43:35 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-x^{2} y^{\prime }-\left (1+x \right )^{2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.570 (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)
 

\[ y = c_{1} \operatorname {HeunT}\left (0, -3, 2 \,3^{{1}/{3}}, \frac {3^{{2}/{3}} x}{3}\right ) {\mathrm e}^{-x}+c_{2} \operatorname {HeunT}\left (0, 3, 2 \,3^{{1}/{3}}, -\frac {3^{{2}/{3}} x}{3}\right ) {\mathrm e}^{\frac {x \left (x^{2}+3\right )}{3}} \]

✓ Solution by Mathematica

Time used: 0.405 (sec). Leaf size: 44

DSolve[-((1 + x)^2*y[x]) - x^2*D[y[x],x] + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{\frac {x^3}{3}+x} \left (c_2 \int _1^xe^{-\frac {1}{3} K[1] \left (K[1]^2+6\right )}dK[1]+c_1\right ) \]