60.3.246 problem 1251

Internal problem ID [11256]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1251
Date solved : Monday, January 27, 2025 at 11:00:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 20

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

\[ y = c_{2} \left (x -1\right ) \ln \left (x \right )-4 c_{2} +c_{1} \left (x -1\right ) \]

✓ Solution by Mathematica

Time used: 0.456 (sec). Leaf size: 112

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

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