60.3.237 problem 1242

Internal problem ID [11247]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1242
Date solved : Monday, January 27, 2025 at 11:00:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-1\right ) y^{\prime \prime }-\left (1+3 x \right ) y^{\prime }-\left (x^{2}-x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 41

dsolve((x^2-1)*diff(diff(y(x),x),x)-(3*x+1)*diff(y(x),x)-(x^2-x)*y(x)=0,y(x), singsol=all)
 

\[ y = {\mathrm e}^{-x -2} c_{2} \left (x +1\right )^{2} \operatorname {Ei}_{1}\left (-2 x -2\right )+c_{1} {\mathrm e}^{-x} \left (x +1\right )^{2}+2 \,{\mathrm e}^{x} c_{2} \]

✓ Solution by Mathematica

Time used: 0.254 (sec). Leaf size: 109

DSolve[(x - x^2)*y[x] - (1 + 3*x)*D[y[x],x] + (-1 + x^2)*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

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