Internal problem ID [8845]
Internal file name [OUTPUT/7780_Monday_June_06_2022_12_18_16_AM_36835657/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 511.
ODE order: 1.
ODE degree: 2.
The type(s) of ODE detected by this program : "first_order_ode_lie_symmetry_calculated"
Maple gives the following as the ode type
[[_1st_order, _with_linear_symmetries]]
\[ \boxed {\left (a^{2} \sqrt {y^{2}+x^{2}}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+a^{2} \sqrt {y^{2}+x^{2}}-y^{2}=0} \] Solving the given ode for \(y^{\prime }\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=-\frac {x y-\sqrt {-y^{2} a^{4}-a^{4} x^{2}+\sqrt {y^{2}+x^{2}}\, y^{2} a^{2}+\sqrt {y^{2}+x^{2}}\, a^{2} x^{2}}}{a^{2} \sqrt {y^{2}+x^{2}}-x^{2}} \tag {1} \\ y^{\prime }&=-\frac {x y+\sqrt {-y^{2} a^{4}-a^{4} x^{2}+\sqrt {y^{2}+x^{2}}\, y^{2} a^{2}+\sqrt {y^{2}+x^{2}}\, a^{2} x^{2}}}{a^{2} \sqrt {y^{2}+x^{2}}-x^{2}} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Writing the ode as \begin {align*} y^{\prime }&=-\frac {x y -\sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )}}{a^{2} \sqrt {x^{2}+y^{2}}-x^{2}}\\ y^{\prime }&= \omega \left ( x,y\right ) \end {align*}
The condition of Lie symmetry is the linearized PDE given by \begin {align*} \eta _{x}+\omega \left ( \eta _{y}-\xi _{x}\right ) -\omega ^{2}\xi _{y}-\omega _{x}\xi -\omega _{y}\eta =0\tag {A} \end {align*}
The type of this ode is not in the lookup table. To determine \(\xi ,\eta \) then (A) is solved using ansatz. Making bivariate polynomials of degree 1 to use as anstaz gives \begin{align*} \tag{1E} \xi &= x a_{2}+y a_{3}+a_{1} \\ \tag{2E} \eta &= x b_{2}+y b_{3}+b_{1} \\ \end{align*} Where the unknown coefficients are \[ \{a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}\} \] Substituting equations (1E,2E) and \(\omega \) into (A) gives \begin{equation} \tag{5E} b_{2}-\frac {\left (x y -\sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )}\right ) \left (b_{3}-a_{2}\right )}{a^{2} \sqrt {x^{2}+y^{2}}-x^{2}}-\frac {\left (x y -\sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )}\right )^{2} a_{3}}{\left (a^{2} \sqrt {x^{2}+y^{2}}-x^{2}\right )^{2}}-\left (-\frac {y -\frac {2 a^{2} x \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )+a^{2} \sqrt {x^{2}+y^{2}}\, x}{2 \sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )}}}{a^{2} \sqrt {x^{2}+y^{2}}-x^{2}}+\frac {\left (x y -\sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )}\right ) \left (\frac {a^{2} x}{\sqrt {x^{2}+y^{2}}}-2 x \right )}{\left (a^{2} \sqrt {x^{2}+y^{2}}-x^{2}\right )^{2}}\right ) \left (x a_{2}+y a_{3}+a_{1}\right )-\left (-\frac {x -\frac {2 a^{2} y \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )+\sqrt {x^{2}+y^{2}}\, y \,a^{2}}{2 \sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )}}}{a^{2} \sqrt {x^{2}+y^{2}}-x^{2}}+\frac {\left (x y -\sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )}\right ) a^{2} y}{\left (a^{2} \sqrt {x^{2}+y^{2}}-x^{2}\right )^{2} \sqrt {x^{2}+y^{2}}}\right ) \left (x b_{2}+y b_{3}+b_{1}\right ) = 0 \end{equation} Putting the above in normal form gives \[ \text {Expression too large to display} \] Setting the numerator to zero gives \begin{equation} \tag{6E} \text {Expression too large to display} \end{equation} Simplifying the above gives \begin{equation} \tag{6E} \text {Expression too large to display} \end{equation} Since the PDE has radicals, simplifying gives \[ \text {Expression too large to display} \] Looking at the above PDE shows the following are all the terms with \(\{x, y\}\) in them. \[ \left \{x, y, \sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )}, \sqrt {x^{2}+y^{2}}\right \} \] The following substitution is now made to be able to collect on all terms with \(\{x, y\}\) in them \[ \left \{x = v_{1}, y = v_{2}, \sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )} = v_{3}, \sqrt {x^{2}+y^{2}} = v_{4}\right \} \] The above PDE (6E) now becomes \begin{equation} \tag{7E} 2 a^{6} v_{1}^{4} a_{2}+4 a^{6} v_{1}^{2} v_{2}^{2} a_{2}+2 a^{6} v_{2}^{4} a_{2}-2 a^{6} v_{1}^{4} b_{3}-4 a^{6} v_{1}^{2} v_{2}^{2} b_{3}-2 a^{6} v_{2}^{4} b_{3}-3 v_{4} a^{4} v_{1}^{4} a_{2}-3 v_{4} a^{4} v_{1}^{2} v_{2}^{2} a_{2}-2 v_{4} a^{4} v_{2}^{4} a_{2}-3 v_{4} a^{4} v_{1}^{3} v_{2} a_{3}-v_{4} a^{4} v_{1} v_{2}^{3} a_{3}-3 v_{4} a^{4} v_{1}^{3} v_{2} b_{2}-v_{4} a^{4} v_{1} v_{2}^{3} b_{2}+4 v_{4} a^{4} v_{1}^{4} b_{3}+3 v_{4} a^{4} v_{1}^{2} v_{2}^{2} b_{3}+v_{4} a^{4} v_{2}^{4} b_{3}+v_{4} a^{4} v_{1}^{3} a_{1}+3 v_{4} a^{4} v_{1} v_{2}^{2} a_{1}+2 a^{4} v_{1}^{2} v_{4} v_{3} a_{3}+2 a^{4} v_{4} v_{3} v_{2}^{2} a_{3}-3 v_{4} a^{4} v_{1}^{2} v_{2} b_{1}-v_{4} a^{4} v_{2}^{3} b_{1}+2 a^{4} v_{1}^{2} v_{4} v_{3} b_{2}+2 a^{4} v_{4} v_{3} v_{2}^{2} b_{2}+a^{2} v_{1}^{6} a_{2}-a^{2} v_{1}^{4} v_{2}^{2} a_{2}-2 a^{2} v_{1}^{2} v_{2}^{4} a_{2}+3 a^{2} v_{1}^{5} v_{2} a_{3}+3 a^{2} v_{1}^{3} v_{2}^{3} a_{3}+3 a^{2} v_{1}^{5} v_{2} b_{2}+3 a^{2} v_{1}^{3} v_{2}^{3} b_{2}-2 a^{2} v_{1}^{6} b_{3}-a^{2} v_{1}^{4} v_{2}^{2} b_{3}+a^{2} v_{1}^{2} v_{2}^{4} b_{3}-a^{2} v_{1}^{5} a_{1}-5 a^{2} v_{1}^{3} v_{2}^{2} a_{1}-4 a^{2} v_{1} v_{2}^{4} a_{1}+2 v_{3} a^{2} v_{1}^{3} v_{2} a_{2}+4 v_{3} a^{2} v_{1} v_{2}^{3} a_{2}-2 a^{2} v_{1}^{4} v_{3} a_{3}-4 v_{3} a^{2} v_{1}^{2} v_{2}^{2} a_{3}+3 a^{2} v_{1}^{4} v_{2} b_{1}+3 a^{2} v_{1}^{2} v_{2}^{3} b_{1}-2 v_{3} a^{2} v_{1}^{4} b_{2}-4 v_{3} a^{2} v_{1}^{2} v_{2}^{2} b_{2}-2 v_{3} a^{2} v_{1} v_{2}^{3} b_{3}+2 v_{3} a^{2} v_{2}^{3} a_{1}+2 v_{3} a^{2} v_{1}^{3} b_{1}+2 v_{3} v_{4} v_{1}^{2} v_{2} a_{1}-2 v_{3} v_{4} v_{1}^{3} b_{1} = 0 \end{equation} Collecting the above on the terms \(v_i\) introduced, and these are \[ \{v_{1}, v_{2}, v_{3}, v_{4}\} \] Equation (7E) now becomes \begin{equation} \tag{8E} \left (-3 a^{4} a_{3}-3 a^{4} b_{2}\right ) v_{1}^{3} v_{2} v_{4}+\left (-4 a^{2} a_{3}-4 a^{2} b_{2}\right ) v_{1}^{2} v_{2}^{2} v_{3}+\left (-3 a^{4} a_{2}+3 a^{4} b_{3}\right ) v_{1}^{2} v_{2}^{2} v_{4}+\left (2 a^{4} a_{3}+2 a^{4} b_{2}\right ) v_{1}^{2} v_{3} v_{4}+\left (4 a^{2} a_{2}-2 a^{2} b_{3}\right ) v_{1} v_{2}^{3} v_{3}+\left (-a^{4} a_{3}-a^{4} b_{2}\right ) v_{1} v_{2}^{3} v_{4}+\left (2 a^{4} a_{3}+2 a^{4} b_{2}\right ) v_{2}^{2} v_{3} v_{4}-a^{2} v_{1}^{5} a_{1}+\left (4 a^{6} a_{2}-4 a^{6} b_{3}\right ) v_{1}^{2} v_{2}^{2}+\left (-2 a^{4} a_{2}+a^{4} b_{3}\right ) v_{2}^{4} v_{4}+\left (3 a^{2} a_{3}+3 a^{2} b_{2}\right ) v_{1}^{5} v_{2}+\left (-a^{2} a_{2}-a^{2} b_{3}\right ) v_{1}^{4} v_{2}^{2}+\left (-2 a^{2} a_{3}-2 a^{2} b_{2}\right ) v_{1}^{4} v_{3}+\left (-3 a^{4} a_{2}+4 a^{4} b_{3}\right ) v_{1}^{4} v_{4}+\left (3 a^{2} a_{3}+3 a^{2} b_{2}\right ) v_{1}^{3} v_{2}^{3}+\left (-2 a^{2} a_{2}+a^{2} b_{3}\right ) v_{1}^{2} v_{2}^{4}-4 a^{2} v_{1} v_{2}^{4} a_{1}+3 a^{2} v_{1}^{4} v_{2} b_{1}-5 a^{2} v_{1}^{3} v_{2}^{2} a_{1}+3 a^{2} v_{1}^{2} v_{2}^{3} b_{1}-v_{4} a^{4} v_{2}^{3} b_{1}+2 v_{3} a^{2} v_{1}^{3} b_{1}+v_{4} a^{4} v_{1}^{3} a_{1}-2 v_{3} v_{4} v_{1}^{3} b_{1}+2 v_{3} a^{2} v_{2}^{3} a_{1}+\left (a^{2} a_{2}-2 a^{2} b_{3}\right ) v_{1}^{6}+\left (2 a^{6} a_{2}-2 a^{6} b_{3}\right ) v_{1}^{4}+\left (2 a^{6} a_{2}-2 a^{6} b_{3}\right ) v_{2}^{4}+3 v_{4} a^{4} v_{1} v_{2}^{2} a_{1}-3 v_{4} a^{4} v_{1}^{2} v_{2} b_{1}+2 v_{3} a^{2} v_{1}^{3} v_{2} a_{2}+2 v_{3} v_{4} v_{1}^{2} v_{2} a_{1} = 0 \end{equation} Setting each coefficients in (8E) to zero gives the following equations to solve \begin {align*} a^{4} a_{1}&=0\\ 2 a_{1}&=0\\ -2 b_{1}&=0\\ -5 a^{2} a_{1}&=0\\ -4 a^{2} a_{1}&=0\\ -a^{2} a_{1}&=0\\ 2 a^{2} a_{1}&=0\\ 2 a^{2} a_{2}&=0\\ 2 a^{2} b_{1}&=0\\ 3 a^{2} b_{1}&=0\\ 3 a^{4} a_{1}&=0\\ -3 a^{4} b_{1}&=0\\ -a^{4} b_{1}&=0\\ -2 a^{2} a_{2}+a^{2} b_{3}&=0\\ -a^{2} a_{2}-a^{2} b_{3}&=0\\ a^{2} a_{2}-2 a^{2} b_{3}&=0\\ 4 a^{2} a_{2}-2 a^{2} b_{3}&=0\\ -3 a^{4} a_{2}+3 a^{4} b_{3}&=0\\ -3 a^{4} a_{2}+4 a^{4} b_{3}&=0\\ -2 a^{4} a_{2}+a^{4} b_{3}&=0\\ 2 a^{6} a_{2}-2 a^{6} b_{3}&=0\\ 4 a^{6} a_{2}-4 a^{6} b_{3}&=0\\ -4 a^{2} a_{3}-4 a^{2} b_{2}&=0\\ -2 a^{2} a_{3}-2 a^{2} b_{2}&=0\\ 3 a^{2} a_{3}+3 a^{2} b_{2}&=0\\ -3 a^{4} a_{3}-3 a^{4} b_{2}&=0\\ -a^{4} a_{3}-a^{4} b_{2}&=0\\ 2 a^{4} a_{3}+2 a^{4} b_{2}&=0 \end {align*}
Solving the above equations for the unknowns gives \begin {align*} a_{1}&=0\\ a_{2}&=0\\ a_{3}&=-b_{2}\\ b_{1}&=0\\ b_{2}&=b_{2}\\ b_{3}&=0 \end {align*}
Substituting the above solution in the anstaz (1E,2E) (using \(1\) as arbitrary value for any unknown in the RHS) gives \begin{align*} \xi &= -y \\ \eta &= x \\ \end{align*} The next step is to determine the canonical coordinates \(R,S\). The canonical coordinates map \(\left ( x,y\right ) \to \left ( R,S \right )\) where \(\left ( R,S \right )\) are the canonical coordinates which make the original ode become a quadrature and hence solved by integration.
The characteristic pde which is used to find the canonical coordinates is \begin {align*} \frac {d x}{\xi } &= \frac {d y}{\eta } = dS \tag {1} \end {align*}
The above comes from the requirements that \(\left ( \xi \frac {\partial }{\partial x} + \eta \frac {\partial }{\partial y}\right ) S(x,y) = 1\). Starting with the first pair of ode’s in (1) gives an ode to solve for the independent variable \(R\) in the canonical coordinates, where \(S(R)\). Therefore \begin {align*} \frac {dy}{dx} &= \frac {\eta }{\xi }\\ &= \frac {x}{-y}\\ &= -\frac {x}{y} \end {align*}
This is easily solved to give \begin {align*} y = \sqrt {-x^{2}+c_{1}} \end {align*}
Where now the coordinate \(R\) is taken as the constant of integration. Hence \begin {align*} R &= x^{2}+y^{2} \end {align*}
Since \(\xi \) depends on \(y\) and \(\eta \) depends on \(x\) then we can use either one to find \(S\). Let us use \begin {align*} dS &= \frac {dx}{\xi } \\ &= \frac {dx}{-y} \end {align*}
But we have now to replace \(y\) in \(\xi \) from its value from the solution of \(\frac {dy}{dx}=\frac {\eta }{\xi }\) found above. This results in \begin {align*} \xi &= -\sqrt {-x^{2}+c_{1}} \end {align*}
Integrating gives \begin {align*} S &= \frac {dx}{-\sqrt {-x^{2}+c_{1}}}\\ &= -\arctan \left (\frac {x}{\sqrt {-x^{2}+c_{1}}}\right ) \end {align*}
Where the constant of integration is set to zero as we just need one solution. Replacing back \(c_{1} = x^{2}+y^{2}\) then the above becomes \begin {align*} S &= -\arctan \left (\frac {x}{\sqrt {y^{2}}}\right ) \end {align*}
Now that \(R,S\) are found, we need to setup the ode in these coordinates. This is done by evaluating \begin {align*} \frac {dS}{dR} &= \frac { S_{x} + \omega (x,y) S_{y} }{ R_{x} + \omega (x,y) R_{y} }\tag {2} \end {align*}
Where in the above \(R_{x},R_{y},S_{x},S_{y}\) are all partial derivatives and \(\omega (x,y)\) is the right hand side of the original ode given by \begin {align*} \omega (x,y) &= -\frac {x y -\sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )}}{a^{2} \sqrt {x^{2}+y^{2}}-x^{2}} \end {align*}
Evaluating all the partial derivatives gives \begin {align*} R_{x} &= 2 x\\ R_{y} &= 2 y\\ S_{x} &= -\frac {y}{x^{2}+y^{2}}\\ S_{y} &= \frac {x}{x^{2}+y^{2}} \end {align*}
Substituting all the above in (2) and simplifying gives the ode in canonical coordinates. \begin {align*} \frac {dS}{dR} &= \frac {-\sqrt {x^{2}+y^{2}}\, y \,a^{2}+\sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )}\, x}{2 \left (a^{2} \sqrt {x^{2}+y^{2}}\, x -x^{3}-x \,y^{2}+\sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )}\, y \right ) \left (x^{2}+y^{2}\right )}\tag {2A} \end {align*}
We now need to express the RHS as function of \(R\) only. This is done by solving for \(x,y\) in terms of \(R,S\) from the result obtained earlier and simplifying. This gives \begin {align*} \frac {dS}{dR} &= -\frac {\left (\sqrt {-a^{2}+\sqrt {R}}\, \sin \left (S \left (R \right )\right )+\cos \left (S \left (R \right )\right ) a \right ) a}{\sqrt {R}\, \left (2 \cos \left (S \left (R \right )\right ) a \sqrt {-a^{2}+\sqrt {R}}\, \sqrt {R}+2 \left (-\sqrt {R}\, a^{2}+R \right ) \sin \left (S \left (R \right )\right )\right )} \end {align*}
The above is a quadrature ode. This is the whole point of Lie symmetry method. It converts an ode, no matter how complicated it is, to one that can be solved by integration when the ode is in the canonical coordiates \(R,S\). Integrating the above gives \begin {align*} S \left (R \right ) = -2 \arctan \left (\frac {\sqrt {-a^{2}+\sqrt {R}}}{a}\right )+c_{1}\tag {4} \end {align*}
To complete the solution, we just need to transform (4) back to \(x,y\) coordinates. This results in \begin {align*} -\arctan \left (\frac {x}{y}\right ) = -2 \arctan \left (\frac {\sqrt {-a^{2}+\sqrt {y^{2}+x^{2}}}}{a}\right )+c_{1} \end {align*}
Which simplifies to \begin {align*} -\arctan \left (\frac {x}{y}\right ) = -2 \arctan \left (\frac {\sqrt {-a^{2}+\sqrt {y^{2}+x^{2}}}}{a}\right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} -\arctan \left (\frac {x}{y}\right ) &= -2 \arctan \left (\frac {\sqrt {-a^{2}+\sqrt {y^{2}+x^{2}}}}{a}\right )+c_{1} \\ \end{align*}
Verification of solutions
\[ -\arctan \left (\frac {x}{y}\right ) = -2 \arctan \left (\frac {\sqrt {-a^{2}+\sqrt {y^{2}+x^{2}}}}{a}\right )+c_{1} \] Verified OK.
Solving equation (2)
Writing the ode as \begin {align*} y^{\prime }&=-\frac {x y +\sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )}}{a^{2} \sqrt {x^{2}+y^{2}}-x^{2}}\\ y^{\prime }&= \omega \left ( x,y\right ) \end {align*}
The condition of Lie symmetry is the linearized PDE given by \begin {align*} \eta _{x}+\omega \left ( \eta _{y}-\xi _{x}\right ) -\omega ^{2}\xi _{y}-\omega _{x}\xi -\omega _{y}\eta =0\tag {A} \end {align*}
The type of this ode is not in the lookup table. To determine \(\xi ,\eta \) then (A) is solved using ansatz. Making bivariate polynomials of degree 1 to use as anstaz gives \begin{align*} \tag{1E} \xi &= x a_{2}+y a_{3}+a_{1} \\ \tag{2E} \eta &= x b_{2}+y b_{3}+b_{1} \\ \end{align*} Where the unknown coefficients are \[ \{a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}\} \] Substituting equations (1E,2E) and \(\omega \) into (A) gives \begin{equation} \tag{5E} b_{2}-\frac {\left (x y +\sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )}\right ) \left (b_{3}-a_{2}\right )}{a^{2} \sqrt {x^{2}+y^{2}}-x^{2}}-\frac {\left (x y +\sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )}\right )^{2} a_{3}}{\left (a^{2} \sqrt {x^{2}+y^{2}}-x^{2}\right )^{2}}-\left (-\frac {y +\frac {2 a^{2} x \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )+a^{2} \sqrt {x^{2}+y^{2}}\, x}{2 \sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )}}}{a^{2} \sqrt {x^{2}+y^{2}}-x^{2}}+\frac {\left (x y +\sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )}\right ) \left (\frac {a^{2} x}{\sqrt {x^{2}+y^{2}}}-2 x \right )}{\left (a^{2} \sqrt {x^{2}+y^{2}}-x^{2}\right )^{2}}\right ) \left (x a_{2}+y a_{3}+a_{1}\right )-\left (-\frac {x +\frac {2 a^{2} y \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )+\sqrt {x^{2}+y^{2}}\, y \,a^{2}}{2 \sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )}}}{a^{2} \sqrt {x^{2}+y^{2}}-x^{2}}+\frac {\left (x y +\sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )}\right ) a^{2} y}{\left (a^{2} \sqrt {x^{2}+y^{2}}-x^{2}\right )^{2} \sqrt {x^{2}+y^{2}}}\right ) \left (x b_{2}+y b_{3}+b_{1}\right ) = 0 \end{equation} Putting the above in normal form gives \[ \text {Expression too large to display} \] Setting the numerator to zero gives \begin{equation} \tag{6E} \text {Expression too large to display} \end{equation} Simplifying the above gives \begin{equation} \tag{6E} \text {Expression too large to display} \end{equation} Since the PDE has radicals, simplifying gives \[ \text {Expression too large to display} \] Looking at the above PDE shows the following are all the terms with \(\{x, y\}\) in them. \[ \left \{x, y, \sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )}, \sqrt {x^{2}+y^{2}}\right \} \] The following substitution is now made to be able to collect on all terms with \(\{x, y\}\) in them \[ \left \{x = v_{1}, y = v_{2}, \sqrt {a^{2} \left (x^{2}+y^{2}\right ) \left (-a^{2}+\sqrt {x^{2}+y^{2}}\right )} = v_{3}, \sqrt {x^{2}+y^{2}} = v_{4}\right \} \] The above PDE (6E) now becomes \begin{equation} \tag{7E} -2 a^{6} v_{1}^{4} a_{2}-4 a^{6} v_{1}^{2} v_{2}^{2} a_{2}-2 a^{6} v_{2}^{4} a_{2}+2 a^{6} v_{1}^{4} b_{3}+4 a^{6} v_{1}^{2} v_{2}^{2} b_{3}+2 a^{6} v_{2}^{4} b_{3}+3 v_{4} a^{4} v_{1}^{4} a_{2}+3 v_{4} a^{4} v_{1}^{2} v_{2}^{2} a_{2}+2 v_{4} a^{4} v_{2}^{4} a_{2}+3 v_{4} a^{4} v_{1}^{3} v_{2} a_{3}+v_{4} a^{4} v_{1} v_{2}^{3} a_{3}+3 v_{4} a^{4} v_{1}^{3} v_{2} b_{2}+v_{4} a^{4} v_{1} v_{2}^{3} b_{2}-4 v_{4} a^{4} v_{1}^{4} b_{3}-3 v_{4} a^{4} v_{1}^{2} v_{2}^{2} b_{3}-v_{4} a^{4} v_{2}^{4} b_{3}-v_{4} a^{4} v_{1}^{3} a_{1}-3 v_{4} a^{4} v_{1} v_{2}^{2} a_{1}+2 a^{4} v_{1}^{2} v_{3} v_{4} a_{3}+2 a^{4} v_{3} v_{4} v_{2}^{2} a_{3}+3 v_{4} a^{4} v_{1}^{2} v_{2} b_{1}+v_{4} a^{4} v_{2}^{3} b_{1}+2 a^{4} v_{1}^{2} v_{3} v_{4} b_{2}+2 a^{4} v_{3} v_{4} v_{2}^{2} b_{2}-a^{2} v_{1}^{6} a_{2}+a^{2} v_{1}^{4} v_{2}^{2} a_{2}+2 a^{2} v_{1}^{2} v_{2}^{4} a_{2}-3 a^{2} v_{1}^{5} v_{2} a_{3}-3 a^{2} v_{1}^{3} v_{2}^{3} a_{3}-3 a^{2} v_{1}^{5} v_{2} b_{2}-3 a^{2} v_{1}^{3} v_{2}^{3} b_{2}+2 a^{2} v_{1}^{6} b_{3}+a^{2} v_{1}^{4} v_{2}^{2} b_{3}-a^{2} v_{1}^{2} v_{2}^{4} b_{3}+a^{2} v_{1}^{5} a_{1}+5 a^{2} v_{1}^{3} v_{2}^{2} a_{1}+4 a^{2} v_{1} v_{2}^{4} a_{1}+2 v_{3} a^{2} v_{1}^{3} v_{2} a_{2}+4 v_{3} a^{2} v_{1} v_{2}^{3} a_{2}-2 a^{2} v_{1}^{4} v_{3} a_{3}-4 v_{3} a^{2} v_{1}^{2} v_{2}^{2} a_{3}-3 a^{2} v_{1}^{4} v_{2} b_{1}-3 a^{2} v_{1}^{2} v_{2}^{3} b_{1}-2 v_{3} a^{2} v_{1}^{4} b_{2}-4 v_{3} a^{2} v_{1}^{2} v_{2}^{2} b_{2}-2 v_{3} a^{2} v_{1} v_{2}^{3} b_{3}+2 v_{3} a^{2} v_{2}^{3} a_{1}+2 v_{3} a^{2} v_{1}^{3} b_{1}+2 v_{3} v_{4} v_{1}^{2} v_{2} a_{1}-2 v_{3} v_{4} v_{1}^{3} b_{1} = 0 \end{equation} Collecting the above on the terms \(v_i\) introduced, and these are \[ \{v_{1}, v_{2}, v_{3}, v_{4}\} \] Equation (7E) now becomes \begin{equation} \tag{8E} \left (3 a^{4} a_{3}+3 a^{4} b_{2}\right ) v_{1}^{3} v_{2} v_{4}+\left (-4 a^{2} a_{3}-4 a^{2} b_{2}\right ) v_{1}^{2} v_{2}^{2} v_{3}+\left (3 a^{4} a_{2}-3 a^{4} b_{3}\right ) v_{1}^{2} v_{2}^{2} v_{4}+\left (2 a^{4} a_{3}+2 a^{4} b_{2}\right ) v_{1}^{2} v_{3} v_{4}+\left (4 a^{2} a_{2}-2 a^{2} b_{3}\right ) v_{1} v_{2}^{3} v_{3}+\left (a^{4} a_{3}+a^{4} b_{2}\right ) v_{1} v_{2}^{3} v_{4}+\left (2 a^{4} a_{3}+2 a^{4} b_{2}\right ) v_{2}^{2} v_{3} v_{4}-3 v_{4} a^{4} v_{1} v_{2}^{2} a_{1}+3 v_{4} a^{4} v_{1}^{2} v_{2} b_{1}+2 v_{3} a^{2} v_{1}^{3} v_{2} a_{2}+2 v_{3} v_{4} v_{1}^{2} v_{2} a_{1}+a^{2} v_{1}^{5} a_{1}+\left (-a^{2} a_{2}+2 a^{2} b_{3}\right ) v_{1}^{6}+\left (-2 a^{6} a_{2}+2 a^{6} b_{3}\right ) v_{1}^{4}+\left (-2 a^{6} a_{2}+2 a^{6} b_{3}\right ) v_{2}^{4}+4 a^{2} v_{1} v_{2}^{4} a_{1}+2 v_{3} a^{2} v_{1}^{3} b_{1}+2 v_{3} a^{2} v_{2}^{3} a_{1}+v_{4} a^{4} v_{2}^{3} b_{1}-3 a^{2} v_{1}^{4} v_{2} b_{1}+5 a^{2} v_{1}^{3} v_{2}^{2} a_{1}-3 a^{2} v_{1}^{2} v_{2}^{3} b_{1}-2 v_{3} v_{4} v_{1}^{3} b_{1}-v_{4} a^{4} v_{1}^{3} a_{1}+\left (-3 a^{2} a_{3}-3 a^{2} b_{2}\right ) v_{1}^{5} v_{2}+\left (a^{2} a_{2}+a^{2} b_{3}\right ) v_{1}^{4} v_{2}^{2}+\left (-2 a^{2} a_{3}-2 a^{2} b_{2}\right ) v_{1}^{4} v_{3}+\left (3 a^{4} a_{2}-4 a^{4} b_{3}\right ) v_{1}^{4} v_{4}+\left (-3 a^{2} a_{3}-3 a^{2} b_{2}\right ) v_{1}^{3} v_{2}^{3}+\left (2 a^{2} a_{2}-a^{2} b_{3}\right ) v_{1}^{2} v_{2}^{4}+\left (-4 a^{6} a_{2}+4 a^{6} b_{3}\right ) v_{1}^{2} v_{2}^{2}+\left (2 a^{4} a_{2}-a^{4} b_{3}\right ) v_{2}^{4} v_{4} = 0 \end{equation} Setting each coefficients in (8E) to zero gives the following equations to solve \begin {align*} a^{2} a_{1}&=0\\ a^{4} b_{1}&=0\\ 2 a_{1}&=0\\ -2 b_{1}&=0\\ 2 a^{2} a_{1}&=0\\ 4 a^{2} a_{1}&=0\\ 5 a^{2} a_{1}&=0\\ 2 a^{2} a_{2}&=0\\ -3 a^{2} b_{1}&=0\\ 2 a^{2} b_{1}&=0\\ -3 a^{4} a_{1}&=0\\ -a^{4} a_{1}&=0\\ 3 a^{4} b_{1}&=0\\ -a^{2} a_{2}+2 a^{2} b_{3}&=0\\ a^{2} a_{2}+a^{2} b_{3}&=0\\ 2 a^{2} a_{2}-a^{2} b_{3}&=0\\ 4 a^{2} a_{2}-2 a^{2} b_{3}&=0\\ 2 a^{4} a_{2}-a^{4} b_{3}&=0\\ 3 a^{4} a_{2}-4 a^{4} b_{3}&=0\\ 3 a^{4} a_{2}-3 a^{4} b_{3}&=0\\ -4 a^{6} a_{2}+4 a^{6} b_{3}&=0\\ -2 a^{6} a_{2}+2 a^{6} b_{3}&=0\\ -4 a^{2} a_{3}-4 a^{2} b_{2}&=0\\ -3 a^{2} a_{3}-3 a^{2} b_{2}&=0\\ -2 a^{2} a_{3}-2 a^{2} b_{2}&=0\\ a^{4} a_{3}+a^{4} b_{2}&=0\\ 2 a^{4} a_{3}+2 a^{4} b_{2}&=0\\ 3 a^{4} a_{3}+3 a^{4} b_{2}&=0 \end {align*}
Solving the above equations for the unknowns gives \begin {align*} a_{1}&=0\\ a_{2}&=0\\ a_{3}&=-b_{2}\\ b_{1}&=0\\ b_{2}&=b_{2}\\ b_{3}&=0 \end {align*}
Substituting the above solution in the anstaz (1E,2E) (using \(1\) as arbitrary value for any unknown in the RHS) gives \begin{align*} \xi &= -y \\ \eta &= x \\ \end{align*} The next step is to determine the canonical coordinates \(R,S\). The canonical coordinates map \(\left ( x,y\right ) \to \left ( R,S \right )\) where \(\left ( R,S \right )\) are the canonical coordinates which make the original ode become a quadrature and hence solved by integration.
The characteristic pde which is used to find the canonical coordinates is \begin {align*} \frac {d x}{\xi } &= \frac {d y}{\eta } = dS \tag {1} \end {align*}
The above comes from the requirements that \(\left ( \xi \frac {\partial }{\partial x} + \eta \frac {\partial }{\partial y}\right ) S(x,y) = 1\). Starting with the first pair of ode’s in (1) gives an ode to solve for the independent variable \(R\) in the canonical coordinates, where \(S(R)\). Unable to determine \(R\). Terminating
Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (a^{2} \sqrt {y^{2}+x^{2}}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+a^{2} \sqrt {y^{2}+x^{2}}-y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=-\frac {x y-\sqrt {-y^{2} a^{4}-a^{4} x^{2}+\sqrt {y^{2}+x^{2}}\, y^{2} a^{2}+\sqrt {y^{2}+x^{2}}\, a^{2} x^{2}}}{a^{2} \sqrt {y^{2}+x^{2}}-x^{2}}, y^{\prime }=-\frac {x y+\sqrt {-y^{2} a^{4}-a^{4} x^{2}+\sqrt {y^{2}+x^{2}}\, y^{2} a^{2}+\sqrt {y^{2}+x^{2}}\, a^{2} x^{2}}}{a^{2} \sqrt {y^{2}+x^{2}}-x^{2}}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {x y-\sqrt {-y^{2} a^{4}-a^{4} x^{2}+\sqrt {y^{2}+x^{2}}\, y^{2} a^{2}+\sqrt {y^{2}+x^{2}}\, a^{2} x^{2}}}{a^{2} \sqrt {y^{2}+x^{2}}-x^{2}} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {x y+\sqrt {-y^{2} a^{4}-a^{4} x^{2}+\sqrt {y^{2}+x^{2}}\, y^{2} a^{2}+\sqrt {y^{2}+x^{2}}\, a^{2} x^{2}}}{a^{2} \sqrt {y^{2}+x^{2}}-x^{2}} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]
Maple trace
`Methods for first order ODEs: -> Solving 1st order ODE of high degree, 1st attempt trying 1st order WeierstrassP solution for high degree ODE trying 1st order WeierstrassPPrime solution for high degree ODE trying 1st order JacobiSN solution for high degree ODE trying 1st order ODE linearizable_by_differentiation trying differential order: 1; missing variables trying simple symmetries for implicit equations Successful isolation of dy/dx: 2 solutions were found. Trying to solve each resulting ODE. *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: trying exact Looking for potential symmetries trying an equivalence to an Abel ODE trying 1st order ODE linearizable_by_differentiation -> Solving 1st order ODE of high degree, Lie methods, 1st trial `, `-> Computing symmetries using: way = 3`[-y, x]
✓ Solution by Maple
Time used: 7.843 (sec). Leaf size: 311
dsolve((a^2*(y(x)^2+x^2)^(1/2)-x^2)*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)+a^2*(y(x)^2+x^2)^(1/2)-y(x)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ \frac {2 \sqrt {-a^{2}+\sqrt {y \left (x \right )^{2}+x^{2}}}\, \sqrt {a^{2} \left (y \left (x \right )^{2}+x^{2}\right )^{2} \left (-a^{2}+\sqrt {y \left (x \right )^{2}+x^{2}}\right )}\, \arctan \left (\frac {\sqrt {-a^{2}+\sqrt {y \left (x \right )^{2}+x^{2}}}}{a}\right )-a \left (y \left (x \right )^{2}+x^{2}\right ) \left (a^{2}-\sqrt {y \left (x \right )^{2}+x^{2}}\right ) \left (c_{1} -\arctan \left (\frac {x}{y \left (x \right )}\right )\right )}{a \left (y \left (x \right )^{2}+x^{2}\right ) \left (a^{2}-\sqrt {y \left (x \right )^{2}+x^{2}}\right )} &= 0 \\ \frac {-2 \sqrt {-a^{2}+\sqrt {y \left (x \right )^{2}+x^{2}}}\, \sqrt {a^{2} \left (y \left (x \right )^{2}+x^{2}\right )^{2} \left (-a^{2}+\sqrt {y \left (x \right )^{2}+x^{2}}\right )}\, \arctan \left (\frac {\sqrt {-a^{2}+\sqrt {y \left (x \right )^{2}+x^{2}}}}{a}\right )-a \left (y \left (x \right )^{2}+x^{2}\right ) \left (a^{2}-\sqrt {y \left (x \right )^{2}+x^{2}}\right ) \left (c_{1} -\arctan \left (\frac {x}{y \left (x \right )}\right )\right )}{a \left (y \left (x \right )^{2}+x^{2}\right ) \left (a^{2}-\sqrt {y \left (x \right )^{2}+x^{2}}\right )} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 42.919 (sec). Leaf size: 229
DSolve[-y[x]^2 + a^2*Sqrt[x^2 + y[x]^2] + 2*x*y[x]*y'[x] + (-x^2 + a^2*Sqrt[x^2 + y[x]^2])*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\arctan \left (\frac {x}{y(x)}\right )-\frac {2 \sqrt {a^2 \left (x^2+y(x)^2\right ) \left (\sqrt {x^2+y(x)^2}-a^2\right )} \arctan \left (\frac {\sqrt {\sqrt {x^2+y(x)^2}-a^2}}{a}\right )}{a \sqrt {x^2+y(x)^2} \sqrt {\sqrt {x^2+y(x)^2}-a^2}}&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {2 \sqrt {a^2 \left (x^2+y(x)^2\right ) \left (\sqrt {x^2+y(x)^2}-a^2\right )} \arctan \left (\frac {\sqrt {\sqrt {x^2+y(x)^2}-a^2}}{a}\right )}{a \sqrt {x^2+y(x)^2} \sqrt {\sqrt {x^2+y(x)^2}-a^2}}+\arctan \left (\frac {x}{y(x)}\right )&=c_1,y(x)\right ] \\ \end{align*}