60.3.405 problem 1411

Internal problem ID [11415]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1411
Date solved : Tuesday, January 28, 2025 at 06:05:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime }&=\frac {y}{{\mathrm e}^{x}+1} \end{align*}

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 27

dsolve(diff(diff(y(x),x),x) = 1/(exp(x)+1)*y(x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.664 (sec). Leaf size: 81

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

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