36.1.4 problem 4

Internal problem ID [6259]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.2, Separable Equations. Exercises. page 46
Problem number : 4
Date solved : Sunday, March 30, 2025 at 10:44:59 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 53

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

\[ \frac {i \sqrt {2}\, {\mathrm e}^{i \sqrt {2}} \operatorname {Ei}_{1}\left (-x +i \sqrt {2}\right )}{4}-\frac {i \sqrt {2}\, {\mathrm e}^{-i \sqrt {2}} \operatorname {Ei}_{1}\left (-x -i \sqrt {2}\right )}{4}+\operatorname {Ei}_{1}\left (y\right )+c_1 = 0 \]

✓ Mathematica. Time used: 0.975 (sec). Leaf size: 81

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

\begin{align*} y(x)\to \text {InverseFunction}[\operatorname {ExpIntegralEi}(-\text {$\#$1})\&]\left [c_1-\frac {i e^{-i \sqrt {2}} \left (e^{2 i \sqrt {2}} \operatorname {ExpIntegralEi}\left (x-i \sqrt {2}\right )-\operatorname {ExpIntegralEi}\left (x+i \sqrt {2}\right )\right )}{2 \sqrt {2}}\right ] \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 1.336 (sec). Leaf size: 20

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

\[ \operatorname {Ei}{\left (e^{i \pi } y{\left (x \right )} \right )} = C_{1} + \int \frac {e^{x}}{x^{2} + 2}\, dx \]