32.6.10 problem Exercise 12.10, page 103

Internal problem ID [5875]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 12, Miscellaneous Methods
Problem number : Exercise 12.10, page 103
Date solved : Monday, January 27, 2025 at 01:22:52 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.085 (sec). Leaf size: 18

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

\[ y \left (x \right ) = \frac {{\mathrm e}^{-x}}{\sqrt {\operatorname {LambertW}\left ({\mathrm e}^{-2 x} c_{1} \right )}} \]

✓ Solution by Mathematica

Time used: 3.331 (sec). Leaf size: 57

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

\begin{align*} y(x)\to -\frac {e^{-x}}{\sqrt {W\left (e^{-2 x+2 c_1}\right )}} \\ y(x)\to \frac {e^{-x}}{\sqrt {W\left (e^{-2 x+2 c_1}\right )}} \\ y(x)\to 0 \\ \end{align*}