2.5.5 problem 7

Internal problem ID [18239]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 32. Problems at page 89
Problem number : 7
Date solved : Monday, December 23, 2024 at 09:18:46 PM
CAS classification : [_quadrature]

Solve

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

Solving for the derivative gives these ODE’s to solve

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

Now each of the above is solved separately.

Solving Eq. (1)

Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).

\begin{align*} \int {dy} &= \int {\frac {1}{\sqrt {x^{2}-1}}\, dx}\\ y &= \ln \left (x +\sqrt {x^{2}-1}\right ) + c_1 \end{align*}

Solving Eq. (2)

Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).

\begin{align*} \int {dy} &= \int {-\frac {1}{\sqrt {x^{2}-1}}\, dx}\\ y &= -\ln \left (x +\sqrt {x^{2}-1}\right ) + c_2 \end{align*}

Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x^{2}-1\right ) {y^{\prime }}^{2}=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {1}{\sqrt {x^{2}-1}}, y^{\prime }=-\frac {1}{\sqrt {x^{2}-1}}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {1}{\sqrt {x^{2}-1}} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {1}{\sqrt {x^{2}-1}}d x +\textit {\_C1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\ln \left (x +\sqrt {x^{2}-1}\right )+\textit {\_C1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\ln \left (x +\sqrt {x^{2}-1}\right )+\textit {\_C1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {1}{\sqrt {x^{2}-1}} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -\frac {1}{\sqrt {x^{2}-1}}d x +\textit {\_C1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\ln \left (x +\sqrt {x^{2}-1}\right )+\textit {\_C1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\ln \left (x +\sqrt {x^{2}-1}\right )+\textit {\_C1} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=-\ln \left (x +\sqrt {x^{2}-1}\right )+\mathit {C1} , y=\ln \left (x +\sqrt {x^{2}-1}\right )+\mathit {C1} \right \} \end {array} \]

Maple trace

`Methods for first order ODEs: 
-> Solving 1st order ODE of high degree, 1st attempt 
trying 1st order WeierstrassP solution for high degree ODE 
trying 1st order WeierstrassPPrime solution for high degree ODE 
trying 1st order JacobiSN solution for high degree ODE 
trying 1st order ODE linearizable_by_differentiation 
trying differential order: 1; missing variables 
<- differential order: 1; missing  y(x)  successful`
 

Maple dsolve solution

Solving time : 0.016 (sec)
Leaf size : 33

dsolve((x^2-1)*diff(y(x),x)^2 = 1, 
       y(x),singsol=all)
 

\begin{align*} y &= \ln \left (x +\sqrt {x^{2}-1}\right )+c_1 \\ y &= -\ln \left (x +\sqrt {x^{2}-1}\right )+c_1 \\ \end{align*}

Mathematica DSolve solution

Solving time : 0.02 (sec)
Leaf size : 41

DSolve[{(x^2-1)*D[y[x],x]^2==1,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

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