28.1.4 problem 4

Internal problem ID [4310]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 4
Date solved : Monday, January 27, 2025 at 09:01:21 AM
CAS classification : [_separable]

\begin{align*} x \left (1+y^{2}\right )+\left (2 y+1\right ) {\mathrm e}^{-x} y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 28

dsolve(x*(y(x)^2+1)+(2*y(x)+1)*exp(-x)*diff(y(x),x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = \tan \left (\operatorname {RootOf}\left (x \,{\mathrm e}^{x}-{\mathrm e}^{x}+\ln \left (2\right )+\ln \left (\frac {1}{1+\cos \left (2 \textit {\_Z} \right )}\right )+\textit {\_Z} +c_{1} \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.625 (sec). Leaf size: 43

DSolve[x*(y[x]^2+1)+(2*y[x]+1)*Exp[-x]*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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