Internal problem ID [5800]
Internal file name [OUTPUT/5048_Sunday_June_05_2022_03_19_01_PM_62397227/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 1.2 Homogeneous equations
problems. page 12
Problem number: 48.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[[_homogeneous, `class G`]]
\[ \boxed {\frac {2 x y y^{\prime }}{3}-\sqrt {x^{6}-y^{4}}-y^{2}=0} \]
Solving for \(y'\) gives \begin{align*} \tag{1} y' &= \frac {\frac {3 \sqrt {x^{6}-y^{4}}}{2}+\frac {3 y^{2}}{2}}{x y} \\ \end{align*} Each of the above ode’s is now solved
Solving ode 1
An ode \(y^{\prime }=f(x,y)\) is isobaric if \[ f(t x, t^m y) = t^{m-1} f(x,y)\tag {1} \] Where here \[ f(x,y) = \frac {\frac {3 \sqrt {x^{6}-y^{4}}}{2}+\frac {3 y^{2}}{2}}{x y}\tag {2} \] \(m\) is the order of isobaric. Substituting (2) into (1) and solving for \(m\) gives \[ m = {\frac {3}{2}} \] Since the ode is isobaric of order \(m={\frac {3}{2}}\), then the substitution \begin {align*} y&=x u^m \\ &=u \,x^{\frac {3}{2}} \end {align*}
Converts the ODE to a separable in \(u \left (x \right )\). Performing this substitution gives \[ \frac {\sqrt {x}\, \left (2 x u^{\prime }\left (x \right )+3 u \left (x \right )\right )}{2} = \frac {\frac {3 \sqrt {x^{6} \left (1-u \left (x \right )^{4}\right )}}{2}+\frac {3 x^{3} u \left (x \right )^{2}}{2}}{x^{\frac {5}{2}} u \left (x \right )} \] Or \[ u^{\prime }\left (x \right ) = \frac {3 \sqrt {x^{6} \left (1-u \left (x \right )^{4}\right )}}{2 x^{4} u \left (x \right )} \] Simplifying the above ode, assuming \(x>0\) gives \[ u^{\prime }\left (x \right ) = \frac {3 \sqrt {1-u \left (x \right )^{4}}}{2 x u \left (x \right )} \] Which is now solved as separable in \(u \left (x \right )\). In canonical form the ODE is \begin {align*} u' &= F(x,u)\\ &= f( x) g(u)\\ &= \frac {3 \sqrt {-u^{4}+1}}{2 x u} \end {align*}
Where \(f(x)=\frac {3}{2 x}\) and \(g(u)=\frac {\sqrt {-u^{4}+1}}{u}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {\sqrt {-u^{4}+1}}{u}} \,du &= \frac {3}{2 x} \,d x \\ \int { \frac {1}{\frac {\sqrt {-u^{4}+1}}{u}} \,du} &= \int {\frac {3}{2 x} \,d x} \\ \frac {\arcsin \left (u^{2}\right )}{2}&=\frac {3 \ln \left (x \right )}{2}+c_{1} \\ \end{align*} The solution is \[ \frac {\arcsin \left (u \left (x \right )^{2}\right )}{2}-\frac {3 \ln \left (x \right )}{2}-c_{1} = 0 \] Now \(u \left (x \right )\) in the above solution is replaced back by \(y\) using \(u=\frac {y}{x^{\frac {3}{2}}}\) which results in the solution \[ \frac {\arcsin \left (\frac {y^{2}}{x^{3}}\right )}{2}-\frac {3 \ln \left (x \right )}{2}-c_{1} = 0 \]
The solution(s) found are the following \begin{align*} \tag{1} \frac {\arcsin \left (\frac {y^{2}}{x^{3}}\right )}{2}-\frac {3 \ln \left (x \right )}{2}-c_{1} &= 0 \\ \end{align*}
Verification of solutions
\[
\frac {\arcsin \left (\frac {y^{2}}{x^{3}}\right )}{2}-\frac {3 \ln \left (x \right )}{2}-c_{1} = 0
\] Verified OK. {0 < x}
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {2 x y y^{\prime }}{3}-\sqrt {x^{6}-y^{4}}-y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {3 \left (\sqrt {x^{6}-y^{4}}+y^{2}\right )}{2 x y} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: trying homogeneous G trying an integrating factor from the invariance group <- integrating factor successful <- homogeneous successful`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 102
dsolve(2/3*x*y(x)*diff(y(x),x)=sqrt(x^6-y(x)^4)+y(x)^2,y(x), singsol=all)
\[ -\left (\int _{\textit {\_b}}^{x}\frac {\sqrt {\textit {\_a}^{6}-y \left (x \right )^{4}}+y \left (x \right )^{2}}{\sqrt {\textit {\_a}^{6}-y \left (x \right )^{4}}\, \textit {\_a}}d \textit {\_a} \right )+\frac {2 \left (\int _{}^{y \left (x \right )}\frac {\textit {\_f} \left (3 \sqrt {x^{6}-\textit {\_f}^{4}}\, \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{5}}{\left (\textit {\_a}^{6}-\textit {\_f}^{4}\right )^{\frac {3}{2}}}d \textit {\_a} \right )+1\right )}{\sqrt {x^{6}-\textit {\_f}^{4}}}d \textit {\_f} \right )}{3}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 6.948 (sec). Leaf size: 128
DSolve[2/3*x*y[x]*y'[x]==Sqrt[x^6-y[x]^4]+y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x^{3/2}}{\sqrt [4]{\sec ^2\left (-\frac {\log \left (x^6\right )}{2}+3 c_1\right )}} \\ y(x)\to -\frac {i x^{3/2}}{\sqrt [4]{\sec ^2\left (-\frac {\log \left (x^6\right )}{2}+3 c_1\right )}} \\ y(x)\to \frac {i x^{3/2}}{\sqrt [4]{\sec ^2\left (-\frac {\log \left (x^6\right )}{2}+3 c_1\right )}} \\ y(x)\to \frac {x^{3/2}}{\sqrt [4]{\sec ^2\left (-\frac {\log \left (x^6\right )}{2}+3 c_1\right )}} \\ \end{align*}