30.1.4 problem 4

Internal problem ID [7394]
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 : Tuesday, September 30, 2025 at 04:30:17 PM
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: 0.397 (sec). Leaf size: 54

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}\left [\int _1^{\text {$\#$1}}\frac {e^{-K[1]}}{K[1]}dK[1]\&\right ]\left [\int _1^x\frac {e^{K[2]}}{K[2]^2+2}dK[2]+c_1\right ]\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.820 (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)
 

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