2.2 problem 2

Internal problem ID [4750]
Internal file name [OUTPUT/4243_Sunday_June_05_2022_12_46_28_PM_67312242/index.tex]

Book: Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section: Chapter 8, Ordinary differential equations. Section 2. Separable equations. page 398
Problem number: 2.
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program : "exact", "separable", "first_order_ode_lie_symmetry_lookup"

Maple gives the following as the ode type

[_separable]

\[ \boxed {x \sqrt {1-y^{2}}+y \sqrt {-x^{2}+1}\, y^{\prime }=0} \] With initial conditions \begin {align*} \left [y \left (\frac {1}{2}\right ) = {\frac {1}{2}}\right ] \end {align*}

2.2.1 Existence and uniqueness analysis

This is non linear first order ODE. In canonical form it is written as \begin {align*} y^{\prime } &= f(x,y)\\ &= -\frac {x \sqrt {-y^{2}+1}}{y \sqrt {-x^{2}+1}} \end {align*}

The \(x\) domain of \(f(x,y)\) when \(y={\frac {1}{2}}\) is \[ \{-1

The \(x\) domain of \(\frac {\partial f}{\partial y}\) when \(y={\frac {1}{2}}\) is \[ \{-1

2.2.2 Solving as separable ode

In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\frac {x \sqrt {-y^{2}+1}}{y \sqrt {-x^{2}+1}} \end {align*}

Where \(f(x)=-\frac {x}{\sqrt {-x^{2}+1}}\) and \(g(y)=\frac {\sqrt {-y^{2}+1}}{y}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {\sqrt {-y^{2}+1}}{y}} \,dy &= -\frac {x}{\sqrt {-x^{2}+1}} \,d x \\ \int { \frac {1}{\frac {\sqrt {-y^{2}+1}}{y}} \,dy} &= \int {-\frac {x}{\sqrt {-x^{2}+1}} \,d x} \\ -\sqrt {-y^{2}+1}&=\sqrt {-x^{2}+1}+c_{1} \\ \end{align*} The solution is \[ -\sqrt {1-y^{2}}-\sqrt {-x^{2}+1}-c_{1} = 0 \] Initial conditions are used to solve for \(c_{1}\). Substituting \(x={\frac {1}{2}}\) and \(y={\frac {1}{2}}\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} -\sqrt {3}-c_{1} = 0 \end {align*}

The solutions are \begin {align*} c_{1} = -\sqrt {3} \end {align*}

Trying the constant \begin {align*} c_{1} = -\sqrt {3} \end {align*}

Substituting \(c_{1}\) found above in the general solution gives \begin {align*} -\sqrt {-y^{2}+1}-\sqrt {-x^{2}+1}+\sqrt {3} = 0 \end {align*}

The constant \(c_{1} = -\sqrt {3}\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} -\sqrt {1-y^{2}}-\sqrt {-x^{2}+1}+\sqrt {3} &= 0 \\ \end{align*}

Verification of solutions

\[ -\sqrt {1-y^{2}}-\sqrt {-x^{2}+1}+\sqrt {3} = 0 \] Verified OK.

2.2.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [x \sqrt {1-y^{2}}+y \sqrt {-x^{2}+1}\, y^{\prime }=0, y \left (\frac {1}{2}\right )=\frac {1}{2}\right ] \\ \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} \\ {} & {} & y^{\prime }=-\frac {x \sqrt {1-y^{2}}}{y \sqrt {-x^{2}+1}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime } y}{\sqrt {1-y^{2}}}=-\frac {x}{\sqrt {-x^{2}+1}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime } y}{\sqrt {1-y^{2}}}d x =\int -\frac {x}{\sqrt {-x^{2}+1}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\sqrt {1-y^{2}}=-\frac {\left (x -1\right ) \left (x +1\right )}{\sqrt {-x^{2}+1}}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\sqrt {-2 c_{1} \sqrt {-x^{2}+1}-c_{1}^{2}+x^{2}}, y=-\sqrt {-2 c_{1} \sqrt {-x^{2}+1}-c_{1}^{2}+x^{2}}\right \} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (\frac {1}{2}\right )=\frac {1}{2} \\ {} & {} & \frac {1}{2}=\sqrt {-\frac {c_{1} \sqrt {3}\, \sqrt {4}}{2}-c_{1}^{2}+\frac {1}{4}} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =\left (0, -\sqrt {3}\right ) \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =\left (0, -\sqrt {3}\right )\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=\mathrm {csgn}\left (x \right ) x \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (\frac {1}{2}\right )=\frac {1}{2} \\ {} & {} & \frac {1}{2}=-\sqrt {-\frac {c_{1} \sqrt {3}\, \sqrt {4}}{2}-c_{1}^{2}+\frac {1}{4}} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =\left (\right ) \\ \bullet & {} & \textrm {Solution does not satisfy initial condition}\hspace {3pt} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=\mathrm {csgn}\left (x \right ) x \end {array} \]

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
<- separable successful`
 

✓ Solution by Maple

Time used: 0.36 (sec). Leaf size: 26

dsolve([x*sqrt(1-y(x)^2)+y(x)*sqrt(1-x^2)*diff(y(x),x)=0,y(1/2) = 1/2],y(x), singsol=all)
 

\[ y \left (x \right ) = \sqrt {2 \sqrt {3}\, \sqrt {-x^{2}+1}+x^{2}-3} \]

✓ Solution by Mathematica

Time used: 3.578 (sec). Leaf size: 38

DSolve[{x*Sqrt[1-y[x]^2]+y[x]*Sqrt[1-x^2]*y'[x]==0,{y[1/2]==1/2}},y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \sqrt {x^2} \\ y(x)\to \sqrt {x^2+2 \sqrt {3-3 x^2}-3} \\ \end{align*}