89.24.11 problem 11

Internal problem ID [24848]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 11. Variation of parameters and other methods. Exercises at page 171
Problem number : 11
Date solved : Thursday, October 02, 2025 at 10:48:37 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*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.177 (sec). Leaf size: 24

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

\begin{align*} y(x)&\to e^x \left (\log \left (e^x+1\right )+(-1+c_2) x+c_1\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 = {} 
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