64.12.14 problem 14

Internal problem ID [13447]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.4. Variation of parameters. Exercises page 162
Problem number : 14
Date solved : Monday, March 31, 2025 at 07:58:08 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=\frac {1}{1+{\mathrm e}^{2 x}} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 32

ode:=diff(diff(y(x),x),x)+3*diff(y(x),x)+2*y(x) = 1/(1+exp(2*x)); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 45

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

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

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-2*y(x)*exp(2*x) - 2*y(x) - exp(2*x)*Derivative(y(x), (x, 2)) - Derivative(y(x), (x, 2)) + 1)/(3*(exp(2*x) + 1)) cannot be solved by the factorable group method