Internal problem ID [5717]
Internal file name [OUTPUT/4965_Sunday_June_05_2022_03_15_23_PM_88416005/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: 4.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {y^{\prime } x -\sqrt {1-y^{2}}=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {\sqrt {-y^{2}+1}}{x} \end {align*}
Where \(f(x)=\frac {1}{x}\) and \(g(y)=\sqrt {-y^{2}+1}\). Integrating both sides gives \begin{align*} \frac {1}{\sqrt {-y^{2}+1}} \,dy &= \frac {1}{x} \,d x \\ \int { \frac {1}{\sqrt {-y^{2}+1}} \,dy} &= \int {\frac {1}{x} \,d x} \\ \arcsin \left (y \right )&=\ln \left (x \right )+c_{1} \\ \end{align*} Which results in \begin{align*} y &= \sin \left (\ln \left (x \right )+c_{1} \right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \sin \left (\ln \left (x \right )+c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ y = \sin \left (\ln \left (x \right )+c_{1} \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } x -\sqrt {1-y^{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\sqrt {1-y^{2}}}{x} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\sqrt {1-y^{2}}}=\frac {1}{x} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\sqrt {1-y^{2}}}d x =\int \frac {1}{x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \arcsin \left (y\right )=\ln \left (x \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\sin \left (\ln \left (x \right )+c_{1} \right ) \end {array} \]
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: 9
dsolve(x*diff(y(x),x)=sqrt(1-y(x)^2),y(x), singsol=all)
\[ y \left (x \right ) = \sin \left (\ln \left (x \right )+c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.219 (sec). Leaf size: 29
DSolve[x*y'[x]==Sqrt[1-y[x]^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \cos (\log (x)+c_1) \\ y(x)\to -1 \\ y(x)\to 1 \\ y(x)\to \text {Interval}[\{-1,1\}] \\ \end{align*}