62.31.1 problem Ex 1

Internal problem ID [12894]
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 : Wednesday, March 05, 2025 at 08:50:45 PM
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: 0.013 (sec). Leaf size: 44

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);
 

\[ y = {\mathrm e}^{\frac {x}{2}-{\mathrm e}^{x}} \sinh \left (\frac {x}{2}\right ) c_{2} +{\mathrm e}^{\frac {x}{2}-{\mathrm e}^{x}} \cosh \left (\frac {x}{2}\right ) c_{1} +{\mathrm e}^{2 x}-4 \,{\mathrm e}^{x}+6 \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 64

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]
 

\[ y(x)\to e^{-e^x} \left (\int _1^x-e^{4 K[1]+e^{K[1]}}dK[1]+e^x \int _1^xe^{3 K[2]+e^{K[2]}}dK[2]+c_2 e^x+c_1\right ) \]

✗ 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