Internal problem ID [5731]
Internal file name [OUTPUT/4979_Sunday_June_05_2022_03_15_48_PM_32759713/index.tex]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.1 Separable equations problems.
page 7
Problem number: 18.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {\frac {y^{\prime }}{\sqrt {1-y^{2}}}=-\frac {1}{\sqrt {-x^{2}+1}}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\frac {\sqrt {-y^{2}+1}}{\sqrt {-x^{2}+1}} \end {align*}
Where \(f(x)=-\frac {1}{\sqrt {-x^{2}+1}}\) and \(g(y)=\sqrt {-y^{2}+1}\). Integrating both sides gives \begin{align*} \frac {1}{\sqrt {-y^{2}+1}} \,dy &= -\frac {1}{\sqrt {-x^{2}+1}} \,d x \\ \int { \frac {1}{\sqrt {-y^{2}+1}} \,dy} &= \int {-\frac {1}{\sqrt {-x^{2}+1}} \,d x} \\ \arcsin \left (y \right )&=\frac {\sqrt {-\left (x -1\right )^{2}-2 x +2}}{2}-\arcsin \left (x \right )-\frac {\sqrt {-\left (1+x \right )^{2}+2 x +2}}{2}+c_{1} \\ \end{align*} Which results in \begin{align*} y &= \sin \left (-\arcsin \left (x \right )+c_{1} \right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \sin \left (-\arcsin \left (x \right )+c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ y = \sin \left (-\arcsin \left (x \right )+c_{1} \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\sqrt {1-y^{2}}}=-\frac {1}{\sqrt {-x^{2}+1}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\sqrt {1-y^{2}}}d x =\int -\frac {1}{\sqrt {-x^{2}+1}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \arcsin \left (y\right )=-\arcsin \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\sin \left (-\arcsin \left (x \right )+c_{1} \right ) \end {array} \]
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: 0.016 (sec). Leaf size: 11
dsolve(1/sqrt(1-x^2)+diff(y(x),x)/sqrt(1-y(x)^2)=0,y(x), singsol=all)
\[ y \left (x \right ) = -\sin \left (\arcsin \left (x \right )+c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.288 (sec). Leaf size: 37
DSolve[1/Sqrt[1-x^2]+y'[x]/Sqrt[1-y[x]^2]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \cos \left (2 \arctan \left (\frac {\sqrt {1-x^2}}{x+1}\right )+c_1\right ) \\ y(x)\to \text {Interval}[\{-1,1\}] \\ \end{align*}