1.3.32 problem 32

Internal problem ID [72]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 1. First order differential equations. Section 1.4 (separable equations). Problems at page 43
Problem number : 32
Date solved : Tuesday, September 30, 2025 at 03:41:34 AM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=y \sqrt {y^{2}-1} \end{align*}

With initial conditions

\begin{align*} y \left (a \right )&=b \\ \end{align*}

✓ Maple. Time used: 0.364 (sec). Leaf size: 28

ode:=diff(y(x),x) = y(x)*(y(x)^2-1)^(1/2); 
ic:=[y(a) = b]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \operatorname {RootOf}\left (-x -\arctan \left (\frac {1}{\sqrt {\textit {\_Z}^{2}-1}}\right )+a +\arctan \left (\frac {1}{\sqrt {b^{2}-1}}\right )\right ) \]

✓ Mathematica. Time used: 0.993 (sec). Leaf size: 113

ode=D[y[x],x]== y[x]*Sqrt[y[x]^2-1]; 
ic={y[a]==b}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\sqrt {\sec ^2\left (a-\arctan \left (\sqrt {b^2-1}\right )-x\right )}\\ y(x)&\to \sqrt {\sec ^2\left (a-\arctan \left (\sqrt {b^2-1}\right )-x\right )}\\ y(x)&\to -\sqrt {\sec ^2\left (a+\arctan \left (\sqrt {b^2-1}\right )-x\right )}\\ y(x)&\to \sqrt {\sec ^2\left (a+\arctan \left (\sqrt {b^2-1}\right )-x\right )} \end{align*}

✓ Sympy. Time used: 0.330 (sec). Leaf size: 44

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

\[ \begin {cases} i \operatorname {acosh}{\left (\frac {1}{y{\left (x \right )}} \right )} & \text {for}\: \frac {1}{\left |{y^{2}{\left (x \right )}}\right |} > 1 \\- \operatorname {asin}{\left (\frac {1}{y{\left (x \right )}} \right )} & \text {otherwise} \end {cases} = - a + x + \begin {cases} i \operatorname {acosh}{\left (\frac {1}{b} \right )} & \text {for}\: \frac {1}{\left |{b^{2}}\right |} > 1 \\- \operatorname {asin}{\left (\frac {1}{b} \right )} & \text {otherwise} \end {cases} \]