56.31.1 problem Ex 1

Internal problem ID [14254]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VIII, Linear differential equations of the second order. Article 54. Change of independent variable. Page 127
Problem number : Ex 1
Date solved : Thursday, October 02, 2025 at 09:27:15 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\left (2 \,{\mathrm e}^{x}-1\right ) y^{\prime }+{\mathrm e}^{2 x} y&={\mathrm e}^{4 x} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 38

ode:=diff(diff(y(x),x),x)+(2*exp(x)-1)*diff(y(x),x)+exp(2*x)*y(x) = exp(4*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (c_1 +c_2 \right ) {\mathrm e}^{x -{\mathrm e}^{x}}}{2}+\frac {\left (c_1 -c_2 \right ) {\mathrm e}^{-{\mathrm e}^{x}}}{2}-4 \,{\mathrm e}^{x}+{\mathrm e}^{2 x}+6 \]

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 39

ode=D[y[x],{x,2}]+(2*Exp[x]-1)*D[y[x],x]+Exp[2*x]*y[x]==Exp[4*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -4 e^x+e^{2 x}+c_1 e^{-e^x}+c_2 e^{x-e^x}+6 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((2*exp(x) - 1)*Derivative(y(x), x) + y(x)*exp(2*x) - exp(4*x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-y(x)*exp(2*x) + exp(4*x) - Derivative(y(x), (x, 2)))/(2*exp(x) - 1) cannot be solved by the factorable group method