Internal problem ID [5765]
Internal file name [OUTPUT/5013_Sunday_June_05_2022_03_17_19_PM_71187612/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: 17.
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 A`], _rational, _dAlembert]
\[ \boxed {\frac {1}{x^{2}-x y+y^{2}}-\frac {y^{\prime }}{2 y^{2}-x y}=0} \]
In canonical form, the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {y \left (-x +2 y \right )}{x^{2}-x y +y^{2}}\tag {1} \end {align*}
An ode of the form \(y' = \frac {M(x,y)}{N(x,y)}\) is called homogeneous if the functions \(M(x,y)\) and \(N(x,y)\) are both homogeneous functions and of the same order. Recall that a function \(f(x,y)\) is homogeneous of order \(n\) if \[ f(t^n x, t^n y)= t^n f(x,y) \] In this case, it can be seen that both \(M=-y \left (x -2 y \right )\) and \(N=x^{2}-x y +y^{2}\) are both homogeneous and of the same order \(n=2\). Therefore this is a homogeneous ode. Since this ode is homogeneous, it is converted to separable ODE using the substitution \(u=\frac {y}{x}\), or \(y=ux\). Hence \[ \frac { \mathop {\mathrm {d}y}}{\mathop {\mathrm {d}x}}= \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}x}}x + u \] Applying the transformation \(y=ux\) to the above ODE in (1) gives \begin {align*} \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}x}}x + u &= \frac {2 u^{2}-u}{u^{2}-u +1}\\ \frac { \mathop {\mathrm {d}u}}{\mathop {\mathrm {d}x}} &= \frac {\frac {2 u \left (x \right )^{2}-u \left (x \right )}{u \left (x \right )^{2}-u \left (x \right )+1}-u \left (x \right )}{x} \end {align*}
Or \[ u^{\prime }\left (x \right )-\frac {\frac {2 u \left (x \right )^{2}-u \left (x \right )}{u \left (x \right )^{2}-u \left (x \right )+1}-u \left (x \right )}{x} = 0 \] Or \[ u^{\prime }\left (x \right ) u \left (x \right )^{2} x -u^{\prime }\left (x \right ) u \left (x \right ) x +u \left (x \right )^{3}+u^{\prime }\left (x \right ) x -3 u \left (x \right )^{2}+2 u \left (x \right ) = 0 \] Or \[ x \left (u \left (x \right )^{2}-u \left (x \right )+1\right ) u^{\prime }\left (x \right )+u \left (x \right )^{3}-3 u \left (x \right )^{2}+2 u \left (x \right ) = 0 \] Which is now solved as separable in \(u \left (x \right )\). Which is now solved in \(u \left (x \right )\). In canonical form the ODE is \begin {align*} u' &= F(x,u)\\ &= f( x) g(u)\\ &= -\frac {u \left (u^{2}-3 u +2\right )}{x \left (u^{2}-u +1\right )} \end {align*}
Where \(f(x)=-\frac {1}{x}\) and \(g(u)=\frac {u \left (u^{2}-3 u +2\right )}{u^{2}-u +1}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {u \left (u^{2}-3 u +2\right )}{u^{2}-u +1}} \,du &= -\frac {1}{x} \,d x \\ \int { \frac {1}{\frac {u \left (u^{2}-3 u +2\right )}{u^{2}-u +1}} \,du} &= \int {-\frac {1}{x} \,d x} \\ -\ln \left (u -1\right )+\frac {\ln \left (u \right )}{2}+\frac {3 \ln \left (u -2\right )}{2}&=-\ln \left (x \right )+c_{2} \\ \end{align*} Raising both side to exponential gives \begin {align*} {\mathrm e}^{-\ln \left (u -1\right )+\frac {\ln \left (u \right )}{2}+\frac {3 \ln \left (u -2\right )}{2}} &= {\mathrm e}^{-\ln \left (x \right )+c_{2}} \end {align*}
Which simplifies to \begin {align*} \frac {\sqrt {u}\, \left (u -2\right )^{\frac {3}{2}}}{u -1} &= \frac {c_{3}}{x} \end {align*}
Now \(u\) in the above solution is replaced back by \(y\) using \(u=\frac {y}{x}\) which results in the solution \[ y = \frac {c_{3}^{2} {\left (\operatorname {RootOf}\left (x^{2} \textit {\_Z}^{8}+2 x^{2} \textit {\_Z}^{6}-\textit {\_Z}^{4} c_{3}^{2}-2 \textit {\_Z}^{2} c_{3}^{2}-c_{3}^{2}\right )^{2}+1\right )}^{2}}{x \operatorname {RootOf}\left (x^{2} \textit {\_Z}^{8}+2 x^{2} \textit {\_Z}^{6}-\textit {\_Z}^{4} c_{3}^{2}-2 \textit {\_Z}^{2} c_{3}^{2}-c_{3}^{2}\right )^{6}} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{3}^{2} {\left (\operatorname {RootOf}\left (x^{2} \textit {\_Z}^{8}+2 x^{2} \textit {\_Z}^{6}-\textit {\_Z}^{4} c_{3}^{2}-2 \textit {\_Z}^{2} c_{3}^{2}-c_{3}^{2}\right )^{2}+1\right )}^{2}}{x \operatorname {RootOf}\left (x^{2} \textit {\_Z}^{8}+2 x^{2} \textit {\_Z}^{6}-\textit {\_Z}^{4} c_{3}^{2}-2 \textit {\_Z}^{2} c_{3}^{2}-c_{3}^{2}\right )^{6}} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{3}^{2} {\left (\operatorname {RootOf}\left (x^{2} \textit {\_Z}^{8}+2 x^{2} \textit {\_Z}^{6}-\textit {\_Z}^{4} c_{3}^{2}-2 \textit {\_Z}^{2} c_{3}^{2}-c_{3}^{2}\right )^{2}+1\right )}^{2}}{x \operatorname {RootOf}\left (x^{2} \textit {\_Z}^{8}+2 x^{2} \textit {\_Z}^{6}-\textit {\_Z}^{4} c_{3}^{2}-2 \textit {\_Z}^{2} c_{3}^{2}-c_{3}^{2}\right )^{6}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {1}{x^{2}-x y+y^{2}}-\frac {y^{\prime }}{2 y^{2}-x y}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {2 y^{2}-x y}{x^{2}-x y+y^{2}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying homogeneous D <- homogeneous successful`
✓ Solution by Maple
Time used: 5.532 (sec). Leaf size: 40
dsolve(1/(x^2-x*y(x)+y(x)^2)=1/(2*y(x)^2-x*y(x))*diff(y(x),x),y(x), singsol=all)
\[ y \left (x \right ) = \left (\operatorname {RootOf}\left (\textit {\_Z}^{8} c_{1} x^{2}+2 \textit {\_Z}^{6} c_{1} x^{2}-\textit {\_Z}^{4}-2 \textit {\_Z}^{2}-1\right )^{2}+2\right ) x \]
✓ Solution by Mathematica
Time used: 60.201 (sec). Leaf size: 1805
DSolve[1/(x^2-x*y[x]+y[x]^2)==1/(2*y[x]^2-x*y[x])*y'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{6} \left (-\sqrt {3} \sqrt {\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}+\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+3 x^2-2 e^{2 c_1}}-\sqrt {3} \sqrt {-\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}-\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+6 x^2+\frac {6 \sqrt {3} x \left (x^2+e^{2 c_1}\right )}{\sqrt {\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}+\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+3 x^2-2 e^{2 c_1}}}-4 e^{2 c_1}}+9 x\right ) \\ y(x)\to \frac {1}{6} \left (-\sqrt {3} \sqrt {\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}+\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+3 x^2-2 e^{2 c_1}}+\sqrt {3} \sqrt {-\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}-\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+6 x^2+\frac {6 \sqrt {3} x \left (x^2+e^{2 c_1}\right )}{\sqrt {\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}+\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+3 x^2-2 e^{2 c_1}}}-4 e^{2 c_1}}+9 x\right ) \\ y(x)\to \frac {1}{6} \left (\sqrt {3} \sqrt {\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}+\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+3 x^2-2 e^{2 c_1}}-\sqrt {3} \sqrt {-\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}-\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+6 x^2-\frac {6 \sqrt {3} x \left (x^2+e^{2 c_1}\right )}{\sqrt {\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}+\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+3 x^2-2 e^{2 c_1}}}-4 e^{2 c_1}}+9 x\right ) \\ y(x)\to \frac {1}{6} \left (\sqrt {3} \sqrt {\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}+\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+3 x^2-2 e^{2 c_1}}+\sqrt {3} \sqrt {-\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}-\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+6 x^2-\frac {6 \sqrt {3} x \left (x^2+e^{2 c_1}\right )}{\sqrt {\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}+\frac {e^{4 c_1}}{\sqrt [3]{54 e^{2 c_1} x^4+6 \sqrt {3} \sqrt {e^{4 c_1} x^4 \left (27 x^4+e^{4 c_1}\right )}+e^{6 c_1}}}+3 x^2-2 e^{2 c_1}}}-4 e^{2 c_1}}+9 x\right ) \\ \end{align*}