70.4.22 problem 28

Internal problem ID [18703]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.6 (Exact equations and integrating factors). Problems at page 100
Problem number : 28
Date solved : Thursday, October 02, 2025 at 03:21:59 PM
CAS classification : [[_1st_order, _with_exponential_symmetries]]

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

✓ Maple. Time used: 0.027 (sec). Leaf size: 24

ode:=y(x)+(2*x*y(x)-exp(-2*y(x)))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{\operatorname {RootOf}\left ({\mathrm e}^{-2 \,{\mathrm e}^{\textit {\_Z}}} c_1 +\textit {\_Z} \,{\mathrm e}^{-2 \,{\mathrm e}^{\textit {\_Z}}}-x \right )} \]

✓ Mathematica. Time used: 0.168 (sec). Leaf size: 25

ode=y[x]+(2*x*y[x]-Exp[-2*y[x]])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [x=e^{-2 y(x)} \log (y(x))+c_1 e^{-2 y(x)},y(x)\right ] \]

✓ Sympy. Time used: 0.583 (sec). Leaf size: 15

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

\[ C_{1} + x e^{2 y{\left (x \right )}} - \log {\left (y{\left (x \right )} \right )} = 0 \]