40.10.5 problem 14

Internal problem ID [6727]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 15. Linear equations with constant coefficients (Variation of parameters). Supplemetary problems. Page 98
Problem number : 14
Date solved : Sunday, March 30, 2025 at 11:20:46 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 41

ode:=diff(diff(y(x),x),x)-y(x) = 1/(1+exp(-x))^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {{\mathrm e}^{x} \left (-4 c_1 -1\right )}{4}-\frac {{\mathrm e}^{-x} \left (4 \ln \left (-\frac {1}{1+{\mathrm e}^{x}}\right )+4 \ln \left (2\right )-4 c_2 +1\right )}{4}-1 \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 42

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

\[ y(x)\to \frac {1}{2} e^{-x} \left (-2 e^x+2 \log \left (e^x+1\right )+2 c_1 e^{2 x}+1+2 c_2\right ) \]

✓ Sympy. Time used: 0.454 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x) + Derivative(y(x), (x, 2)) - 1/(1 + exp(-x))**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} - \frac {1}{2 \left (e^{x} + 1\right )}\right ) e^{x} + \left (C_{2} + \log {\left (e^{x} + 1 \right )}\right ) e^{- x} - \frac {1}{2} + \frac {e^{- x}}{2 \left (e^{x} + 1\right )} \]