60.3.257 problem 1262

Internal problem ID [11267]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1262
Date solved : Monday, January 27, 2025 at 11:01:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.822 (sec). Leaf size: 53

dsolve((x+1)^2*diff(diff(y(x),x),x)+(x^2+x-1)*diff(y(x),x)-(x+2)*y(x)=0,y(x), singsol=all)
 

\[ y = \left (c_{1} {\mathrm e}^{-x} \operatorname {HeunD}\left (4, 4, -8, 12, \frac {x}{x +2}\right )+c_{2} \operatorname {HeunD}\left (-4, 4, -8, 12, \frac {x}{x +2}\right ) {\mathrm e}^{\frac {x -1}{2+2 x}}\right ) \left (x +1\right ) \]

✓ Solution by Mathematica

Time used: 0.304 (sec). Leaf size: 105

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

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