38.5.30 problem 30

Internal problem ID [8378]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.2 Separable equations. Exercises 2.2 at page 53
Problem number : 30
Date solved : Tuesday, September 30, 2025 at 05:33:06 PM
CAS classification : [_separable]

\begin{align*} x \sinh \left (y\right ) y^{\prime }&=\cosh \left (y\right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.450 (sec). Leaf size: 48

ode:=x*sinh(y(x))*diff(y(x),x) = cosh(y(x)); 
ic:=[y(1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\begin{align*} y &= \operatorname {arccoth}\left (\frac {\sqrt {x^{4}-x^{2}}}{x^{2}-1}\right ) \\ y &= \operatorname {arccoth}\left (-\frac {\sqrt {x^{4}-x^{2}}}{x^{2}-1}\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.557 (sec). Leaf size: 15

ode=x*Sinh[y[x]]*D[y[x],x]==Cosh[y[x]]; 
ic={y[1]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\text {arccosh}(x)\\ y(x)&\to \text {arccosh}(x) \end{align*}

✓ Sympy. Time used: 0.311 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*sinh(y(x))*Derivative(y(x), x) - cosh(y(x)),0) 
ics = {y(1): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} - \log {\left (x \right )} - \log {\left (\tanh {\left (y{\left (x \right )} \right )} + 1 \right )} = 0 \]