28.2.64 problem 64

Internal problem ID [4507]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 4. Linear Differential Equations. Page 183
Problem number : 64
Date solved : Tuesday, March 04, 2025 at 06:49:31 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 0.325 (sec). Leaf size: 24

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

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

✗ 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) + exp(2*x)*Derivative(y(x), (x, 2)) - exp(2*x) + 2*exp(x)*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 2)))/(2*(exp(2*x) + 2*exp(x) + 1)) cannot be solved by the factorable group method