39.3.2 problem Problem 12.2

Internal problem ID [6528]
Book : Schaums Outline Differential Equations, 4th edition. Bronson and Costa. McGraw Hill 2014
Section : Chapter 12. VARIATION OF PARAMETERS. page 104
Problem number : Problem 12.2
Date solved : Sunday, March 30, 2025 at 11:06:33 AM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 56

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

\[ y = \frac {\left (-2 \,{\mathrm e}^{x}-{\mathrm e}^{2 x}-1\right ) \ln \left (1+{\mathrm e}^{-x}\right )}{2}+\frac {\left (2 \,{\mathrm e}^{x}+1\right ) \ln \left ({\mathrm e}^{-x}\right )}{2}+\frac {{\mathrm e}^{2 x} c_1}{2}+\frac {\left (2 c_2 +1\right ) {\mathrm e}^{x}}{2}+c_3 \]

✓ Mathematica. Time used: 0.178 (sec). Leaf size: 59

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

\[ y(x)\to \frac {1}{2} \left (-2 e^{2 x} \text {arctanh}\left (2 e^x+1\right )-\left (2 e^x+1\right ) \log \left (e^x+1\right )+e^x \left (c_2 e^x+1+2 c_1\right )\right )+c_3 \]

✗ Sympy

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

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