76.1.25 problem 25

Internal problem ID [17253]
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 2.1 (Separable equations). Problems at page 44
Problem number : 25
Date solved : Monday, March 31, 2025 at 03:46:34 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {{\mathrm e}^{-x}-{\mathrm e}^{x}}{3+4 y} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.291 (sec). Leaf size: 29

ode:=diff(y(x),x) = (exp(-x)-exp(x))/(3+4*y(x)); 
ic:=y(0) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = -\frac {3}{4}+\frac {\sqrt {{\mathrm e}^{x} \left (-8 \,{\mathrm e}^{2 x}+65 \,{\mathrm e}^{x}-8\right )}\, {\mathrm e}^{-x}}{4} \]

✓ Mathematica. Time used: 1.34 (sec). Leaf size: 29

ode=D[y[x],x]==(Exp[-x]-Exp[x])/(3+4*y[x]); 
ic={y[0]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{4} \left (\sqrt {-8 e^{-x}-8 e^x+65}-3\right ) \]

✓ Sympy. Time used: 0.769 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (-exp(x) + exp(-x))/(4*y(x) + 3),0) 
ics = {y(0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\sqrt {65 - 16 \cosh {\left (x \right )}}}{4} - \frac {3}{4} \]