Internal problem ID [15419]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 15.5 Linear equations with variable coefficients.
The Lagrange method. Exercises page 148
Problem number: 654.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_y]]
\[ \boxed {y^{\prime \prime }+y^{\prime }=\frac {1}{1+{\mathrm e}^{x}}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 30
dsolve(diff(y(x),x$2)+diff(y(x),x)=1/(1+exp(x)),y(x), singsol=all)
\[ y \left (x \right ) = \left (-{\mathrm e}^{-x}-1\right ) \ln \left (1+{\mathrm e}^{x}\right )-c_{1} {\mathrm e}^{-x}+c_{2} +\ln \left ({\mathrm e}^{x}\right ) \]
✓ Solution by Mathematica
Time used: 0.098 (sec). Leaf size: 33
DSolve[y''[x]+y'[x]==1/(1+Exp[x]),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x-\log \left (e^x+1\right )-e^{-x} \left (\log \left (e^x+1\right )+c_1\right )+c_2 \]