87.18.24 problem 24

Internal problem ID [23665]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 135
Problem number : 24
Date solved : Thursday, October 02, 2025 at 09:44:03 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&=3 \\ y^{\prime }\left (0\right )&={\frac {5}{2}} \\ \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 21

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)+y(x) = exp(2*x)/(exp(x)+1)^2; 
ic:=[y(0) = 3, D(y)(0) = 5/2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = {\mathrm e}^{x} \left (-\ln \left (2\right )+\ln \left ({\mathrm e}^{x}+1\right )-x +3\right ) \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 23

ode=D[y[x],{x,2}]-2*D[y[x],x]+y[x]==Exp[2*x]/(Exp[x]+1)^2; 
ic={y[0]==3,Derivative[1][y][0] ==5/2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -e^x \left (x-\log \left (e^x+1\right )-3+\log (2)\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - exp(2*x)/(exp(x) + 1)**2,0) 
ics = {y(0): 3, Subs(Derivative(y(x), x), x, 0): 5/2} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (y(x)*exp(2*x) + 2*y(x)*exp(x) + y(x) + ex