7.3.32 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 : Friday, February 07, 2025 at 07:47:49 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*}

✓ Solution by Maple

Time used: 0.310 (sec). Leaf size: 28

dsolve([diff(y(x),x)= y(x)*sqrt(y(x)^2-1),y(a) = b],y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.694 (sec). Leaf size: 113

DSolve[{D[y[x],x]== y[x]*Sqrt[y[x]^2-1],{y[a]==b}},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*}