60.3.320 problem 1326

Internal problem ID [11330]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1326
Date solved : Monday, January 27, 2025 at 11:13:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 22

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

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

✓ Solution by Mathematica

Time used: 0.408 (sec). Leaf size: 77

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

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