36.2.24 problem 29

Internal problem ID [6317]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.3, Linear equations. Exercises. page 54
Problem number : 29
Date solved : Sunday, March 30, 2025 at 10:51:15 AM
CAS classification : [[_1st_order, _with_exponential_symmetries]]

\begin{align*} \left ({\mathrm e}^{4 y}+2 x \right ) y^{\prime }-1&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 41

ode:=(exp(4*y(x))+2*x)*diff(y(x),x)-1 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\ln \left (-c_1 -\sqrt {c_1^{2}+2 x}\right )}{2} \\ y &= \frac {\ln \left (-c_1 +\sqrt {c_1^{2}+2 x}\right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.18 (sec). Leaf size: 113

ode=(Exp[4*y[x]]+2*x)*D[y[x],x]-1==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \log \left (-\sqrt {-\sqrt {2 x+c_1{}^2}-c_1}\right ) \\ y(x)\to \frac {1}{2} \log \left (-\sqrt {2 x+c_1{}^2}-c_1\right ) \\ y(x)\to \log \left (-\sqrt {\sqrt {2 x+c_1{}^2}-c_1}\right ) \\ y(x)\to \frac {1}{2} \log \left (\sqrt {2 x+c_1{}^2}-c_1\right ) \\ \end{align*}

✓ Sympy. Time used: 4.647 (sec). Leaf size: 78

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((2*x + exp(4*y(x)))*Derivative(y(x), x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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