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I have a doubt respect to resolution of differential equations, for example if we have the family of circles $x^2+y^2=2cx$, deriving $$2x+2y\frac{dy}{dx}=2c$$, combining $$\frac{dy}{dx}=\frac{y^2-x^2}{2xy}$$ and replacing $\frac{dy}{dx}=-\frac{dx}{dy}$, then we have the differential equation $$\frac{dy}{dx}=\frac{2xy}{x^2-y^2}(*)$$ We cannot find the solution of the last differential equation by separation of variables, but if we use polar coordinates in $x^2+y^2=2cx$ we get $$(r\cos\theta)^2+(r\sin\theta)^2=2c(r\cos\theta)$$ and $r=2c\cos\theta$, then $$\frac{dr}{d\theta}=-2c\sin\theta$$ and then $$\frac{r d\theta}{dr}=-\frac{\cos\theta}{\sin\theta}$$

The solution of the last differential equation is $r=2c\sin\theta$, so the solution of the differential equation (*) is $x^2+y^2=2cy$. Then my question is: why the change of coordinates permit find the solution of the differential equation? This is an accident or exit a theorem about this? And if exits such theorem, what kind of differential equation can be solve by change of coordinates? Thanks.

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Somehow, you have gone from $x^2+y^2=2cx$ to $x^2+y^2=2cy$. – Gerry Myerson Oct 21 '12 at 5:02
I'll add more information. – José Ramírez Oct 21 '12 at 5:07
You have added more information, but you have not resolved the contradiction. In the second line, you have $x^2+y^2=2cx$. Later, you have $x^2+y^2=2cy$. These are different. One of them has an $x$ where the other has a $y$. How can both of them be the solution, when they are manifestly not equal to each other? – Gerry Myerson Oct 21 '12 at 5:23
@GerryMyerson: I think the problem is the step in which $$\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{\mathrm{d}x}{\mathrm{d}y}$$ – robjohn Oct 21 '12 at 10:43
@robjohn, yes, that step looks highly problematical, especially since what has actually been done looks more like $dy/dx=-1/(dy/dx)$. – Gerry Myerson Oct 21 '12 at 11:48
up vote 1 down vote accepted

Solving (*)

We can solve $(\ast)$ without changing to polar coordinates $$ \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{2xy}{x^2-y^2}\tag{1} $$ which can be manipulated to $$ \frac{\mathrm{d}}{\mathrm{d}y}\frac{x^2}{y}=\frac{2x}{y}\frac{\mathrm{d}x}{\mathrm{d}y}-\frac{x^2}{y^2}=-1\tag{2} $$ Integrating $(2)$ yields $$ \frac{x^2}{y}=2c-y\tag{3} $$ For some $c$. Therefore, $$ x^2+y^2=2cy\tag{4} $$

Answer to the Question

A change of coordinate may make the solution more apparent, but the equation should be solvable in either coordinate system.

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The equation $$\frac{dy}{dx}=\frac{2xy}{x^2-y^2}(*)$$ is ok, but you don't answer the main question. – José Ramírez Oct 21 '12 at 19:52
The main question was: "why [does] the change of coordinates permit [us to] find the solution of the differential equation?" My answer was "A change of coordinate may make the solution more apparent, but the equation should be solvable in either coordinate system." – robjohn Oct 21 '12 at 21:25
The original equation $x^2+y^2=2cx$ has nothing to do with $(\ast)$ as written. That is the point which was confusing to me, and possibly to Gerry Myerson. – robjohn Oct 21 '12 at 21:28

You can use a substitution for the Bernoulli DE you obtain when considering $\frac{dx}{dy} = \frac{x^2 - y^2}{2xy}$

Edit: Notice $\frac{dx}{dy} = \frac{x^2 - y^2}{2xy} \Rightarrow x' - \frac{x}{2y} + \frac{y}{2x} = 0$, where $x'$ is obviously with respect to $y$

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Could you be more expecific? That is not the exact form like here – José Ramírez Oct 21 '12 at 4:50
It appears to be for $n=-1$ where $$ P(y)=\frac1{2y}\qquad\text{and}\qquad Q(y)=\frac x2 $$ – robjohn Oct 21 '12 at 10:54
But i know the answer of this, the problem is about this equation $$\frac{dy}{dx}=\frac{2xy}{x^2-y^2}(*)$$ and change of coordinates, you don't answer the main question. – José Ramírez Oct 21 '12 at 19:56

When you replace $\frac{dy}{dx}=-\frac{dx}{dy}$, you get

$$ \frac{dx}{dy}=-\frac{y^2-x^2}{2xy}= -\frac{y}{2x}+\frac{x}{2y} \rightarrow (1)\,. $$

It is easier to solve the differential equation (1). Let

$$u=\frac{x}{y} \implies x=yu \implies \frac{dx}{dy}=u+y\frac{du}{dy} \,. $$

Substituting back in $(1)$ gives,

$$ u+y\frac{du}{dy}= -\frac{1}{2u}+\frac{u}{2}\implies y\frac{du}{dy} =\frac{3u^2-1}{2u}\,. $$

Now, I think you can solve the last ode by the method of separation of variables to find $u$ as a function in $y$, then, substitute back $ u=\frac{x}{y} $.

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I saw this only now ... appears not to have been answered to your satisfaction.

You are not at all changing coordinates!! You are finding orthogonal trajectories of

$$ x^2 + y^2 + 2 c_1 x =0 $$ as

$$ x^2 + y^2 + 2 c_2 y =0. $$

with transformation $ y^{'} $ to $ -1/y^{'} $ on the same coordinate axes, which is the standard procedure to find orthogonal trajectories by a changed differential equation on same coordinate axes of same graph sheet (Red changes to Blue).

$ c_1, c_2 $ are parameters in each set of circles, displacement on axis = radius of circle. I shall not repeat the integration steps.

Their polar equations are

$$ r = 2 c_1 \cos\theta,\;\; r = 2 c_1 \sin \theta $$

The fist one is a set of circles centered on x-axis touching y-axis and passing through origin and the second is the orthogonal set of circles centered on y-axis, also passing through the origin and touching x-axis as shown in the sketch


Incidentally, it is a special case of bipolar system of coordinates and also geodesics of the hyperbolic plane.

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