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 : Monday, January 27, 2025 at 01:50:19 PM
CAS classification : [_separable]

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

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

DSolve[D[y[x],x]==y[x]*Exp[x+y[x]]/(x^2+2),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*}