60.3.372 problem 1378

Internal problem ID [11382]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1378
Date solved : Monday, January 27, 2025 at 11:18:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 48

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

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

✓ Solution by Mathematica

Time used: 0.235 (sec). Leaf size: 90

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

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