Internal problem ID [5732]
Internal file name [OUTPUT/4980_Sunday_June_05_2022_03_15_50_PM_91504895/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: 19.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_separable]
\[ \boxed {2 x \sqrt {1-y^{2}}+y y^{\prime }=0} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= -\frac {2 x \sqrt {-y^{2}+1}}{y} \end {align*}
Where \(f(x)=-2 x\) and \(g(y)=\frac {\sqrt {-y^{2}+1}}{y}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {\sqrt {-y^{2}+1}}{y}} \,dy &= -2 x \,d x \\ \int { \frac {1}{\frac {\sqrt {-y^{2}+1}}{y}} \,dy} &= \int {-2 x \,d x} \\ -\sqrt {-y^{2}+1}&=-x^{2}+c_{1} \\ \end{align*} The solution is \[ -\sqrt {1-y^{2}}+x^{2}-c_{1} = 0 \]
The solution(s) found are the following \begin{align*} \tag{1} -\sqrt {1-y^{2}}+x^{2}-c_{1} &= 0 \\ \end{align*}
Verification of solutions
\[ -\sqrt {1-y^{2}}+x^{2}-c_{1} = 0 \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 x \sqrt {1-y^{2}}+y y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {2 x \sqrt {1-y^{2}}}{y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y y^{\prime }}{\sqrt {1-y^{2}}}=-2 x \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y y^{\prime }}{\sqrt {1-y^{2}}}d x =\int -2 x d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\sqrt {1-y^{2}}=-x^{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\sqrt {-x^{4}+2 c_{1} x^{2}-c_{1}^{2}+1}, y=-\sqrt {-x^{4}+2 c_{1} x^{2}-c_{1}^{2}+1}\right \} \end {array} \]
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: 0.0 (sec). Leaf size: 27
dsolve(2*x*sqrt(1-y(x)^2)+y(x)*diff(y(x),x)=0,y(x), singsol=all)
\[ c_{1} +x^{2}+\frac {\left (y \left (x \right )-1\right ) \left (y \left (x \right )+1\right )}{\sqrt {1-y \left (x \right )^{2}}} = 0 \]
✓ Solution by Mathematica
Time used: 0.288 (sec). Leaf size: 69
DSolve[2*x*Sqrt[1-y[x]^2]+y[x]*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {-x^4+2 c_1 x^2+1-c_1{}^2} \\ y(x)\to \sqrt {-x^4+2 c_1 x^2+1-c_1{}^2} \\ y(x)\to -1 \\ y(x)\to 1 \\ \end{align*}