83.27.2 problem 2

Internal problem ID [19277]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (A) at page 104
Problem number : 2
Date solved : Monday, March 31, 2025 at 07:05:23 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 22

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

\[ y = 1+{\mathrm e}^{-{\mathrm e}^{x}} \left (c_1 \,\operatorname {Ei}_{1}\left (-{\mathrm e}^{x}\right )+c_2 \right ) \]

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 28

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

\[ y(x)\to e^{-e^x} \left (c_1 \operatorname {ExpIntegralEi}\left (e^x\right )+e^{e^x}+c_2\right ) \]

✗ Sympy

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

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