Orthogonal Trajectories Using Polar Coordinates. Correct Calculations, Two Different Answers?

Question

My textbook, George F. Simmons' Differential Equations with Applications and Historical Notes, asks to find the orthogonal trajectory of the family of curves $r = 2Ccos(\theta)$ where C is a parameter. The original equation of the family of curves was $x^2 + y^2 = 2Cx$, but it led to an equation that was as yet unsolvable using the methods taught by the textbook up to that moment.$^*$ To compensate, the textbook switched to polar coordinates and started solving it that way, which was what I have shown in the calculations below.

$*$ For further elaboration, the authors got $\dfrac{dy}{dx} = \dfrac{2xy}{x^2 - y^2}$ from $x^2 + y^2 = 2Cx$. They then said, "Unfortunately, the variables cannot be separated, so without additional techniques for solving differential equations we can go no further in this direction. However, if we use polar coordinates, the equation of the family can be written as $r = 2Ccos(\theta)$". And then they continue with their calculations, as stated below.

The solution that I am getting is different from that of the textbook. All of the (similar) previous problems that I have completed have been correct, so if there are errors in my understanding of this concept, I cannot detect them.

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My Solution

$r = 2Ccos(\theta)$

$\dfrac{dr}{d\theta} = -2Csin(\theta)$

We need to eliminate the arbitrary constant $C$ because we don't just want the orthogonal trajectory for a single curve -- we want the orthogonal trajectories for the entire family of curves; therefore, we want $\dfrac{dr}{d\theta}$ in terms of $r$ and $\theta$.

$\dfrac{r}{2\cos(\theta)} = C$

$\therefore \dfrac{dr}{d\theta} = -2\left( \dfrac{r}{2cos(\theta)} \right) sin(\theta)$

$= \dfrac{-rsin(\theta)}{cos(\theta)}$

The orthogonal trajectories will have a slope which is the negative reciprocal of the slopes of the family of curves:

$\therefore \dfrac{-d\theta}{dr} = \dfrac{-rsin(\theta)}{cos(\theta)}$

$\implies \dfrac{d\theta}{dr} = \dfrac{rsin(\theta)}{cos(\theta)}$

$\implies \dfrac{dr}{d\theta} = \dfrac{cos(\theta)}{rsin(\theta)}$

$\implies dr(r) = \dfrac{cos(\theta)}{sin(\theta)} (d\theta)$

And we can now proceed with separation of variables...

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Textbook's Solution

$r = 2Ccos(\theta)$

$\dfrac{dr}{d\theta} = -2Csin(\theta)$

After eliminating C we arrive at

$\dfrac{rd\theta}{dr} = \dfrac{-cos(\theta)}{sin(\theta)}$

as the differential equation of the given family. Accordingly,

$\dfrac{rd\theta}{dr} = \dfrac{sin(\theta)}{cos(\theta)}$

is the differential equation of the orthogonal trajectories. In this case, the variables can be separated, yielding

$\dfrac{dr}{r} = \dfrac{cos(\theta) d\theta}{sin(\theta)}$

And it then proceeds with integration ...

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I'm wondering if both solutions (mine and the textbook) are correct? Or have I made an error? If I've made an error, I would appreciate it if people could please take the time to carefully explain the reasoning behind it. I have only just begun studying differential equations (chapter 1), so any explanation would have to be very elementary.

Answer 1

Everything is wrong here. Note that if you take a circle, say, $r=2$, then $dr/d\theta=0$. And yet the slope of the curve at $(x,y)$ is $-\dfrac xy=-\cot\theta$. There is, of course, a formula for the slope in terms of $dr/d\theta$, but it is not so simple.

I don't understand the problem with doing this in cartesian coordinates. We start with $x^2+y^2=2Cx$, we find the slope to be $$\frac{dy}{dx} = \frac{C-x}y = \frac{y^2-x^2}{2xy}.$$ The orthogonal trajectories will be found by solving $$\frac{dy}{dx} = \frac{2xy}{x^2-y^2}.$$ This is a homogeneous ordinary differential equation. Substitute $y=ux$, so $$x\frac{du}{dx}+ u = \frac{2u}{1-u^2},$$ and we end up with the separable differential equation $$\frac{1-u^2}{u(1+u^2)}du = \frac{dx}x.$$ This is not so bad. The left-hand side simplifies to $\dfrac 1u - \dfrac{2u}{1+u^2}$, and so we integrate and obtain $$\log u - \log (1+u^2) = \log x + c', \quad\text{i.e.,}\quad \frac u{1+u^2}=cx.$$ Not so surprisingly, this turns into the family of circles $x^2+y^2 = \frac1c y$.

EDIT: With a bit of work, one can show that $$\frac{dr}{d\theta} = r \frac{1+m\tan\theta}{m-\tan\theta}$$ where $m=dy/dx$ is the slope of the curve. One can't miss the resemblance to the addition formula for tangent here! This can be rewritten as $\displaystyle{\frac1r\frac{dr}{d\theta} = \frac1{\tan(\phi-\theta)}}$, where $\tan\phi = m = dy/dx$. Now put in $-1/m$ for $m$ and you'll get the orthogonal trajectory with $$\frac{dr}{d\theta} = -r\frac{m-\tan\theta}{1+m\tan\theta} = -r^2\frac1{\text{original }dr/d\theta}.$$ (Note that this is replacing $\tan\phi$ with $\tan(\phi+\pi/2) = -1/(\tan\phi)$.)

Answer 2

Edited (Not the slope, but the orthogonal direction)

The correct is the one from your textbook. In polars, the perpendicular is got by means $\dfrac{1}{r}\dfrac{\mathrm dr}{\mathrm d\theta}\to-r\dfrac{\mathrm d\theta}{\mathrm dr}$

Added

Suppose the position vector for each curve points with parameter $t$: $(r,\theta)=(r(t),\theta(t))$

It's known the vector tangent to this curve is:

$$\frac{\mathrm d\mathbf v}{\mathrm dt}=\frac{\mathrm dr}{\mathrm dt}\mathbf{\hat r}+\frac{\mathrm d\mathbf\theta}{\mathrm dt}r\mathbf{\hat\theta}$$

An orthogonal vector to this one is:

$$\mathbf v_p=r\frac{\mathrm d\mathbf \theta}{\mathrm dt}\mathbf{\hat r}-\frac{\mathrm dr}{\mathrm dt}\mathbf{\hat\theta}$$

You can now consider eliminating the parameter for the curve:

$$\frac{\mathrm dr/\mathrm dt}{r\mathrm d\mathbf \theta/\mathrm dt}=\frac{\mathrm dr}{r\mathrm d\theta}=f(r,\theta)$$

Now,

$$-\frac{r\mathrm d\theta}{\mathrm dr}=f(r,\theta)$$

is true of the orthogonal curves to the original one.

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$r=2C\cos\theta$, $t=\theta$, $(r(t),\theta(t))=(2C\cos(t),t)$

$$\frac{\mathrm d\mathbf v}{\mathrm dt}=\frac{\mathrm dr}{\mathrm dt}\mathbf{\hat r}+\frac{\mathrm d\mathbf\theta}{\mathrm dt}r\mathbf{\hat\theta}=-2C\sin(t)\mathbf{\hat r}+1·2C\cos(t)\mathbf{\hat\theta}$$

$$\frac{\mathrm dr/\mathrm dt}{r\mathrm d\mathbf \theta/\mathrm dt}=\frac{-2C\sin(t)}{2C\cos(t)}=\frac{-\sin\theta}{\cos\theta}=\frac{\mathrm dr}{r\mathrm d\theta}$$

and

$$\frac{r\mathrm d\theta}{\mathrm dr}=\frac{\sin\theta}{\cos\theta}$$

Answer 3

$$ r= 2C \cos \theta,~~ r'= - 2C \sin \theta~d \theta $$

Dividing

$$\frac{r}{r'}=- \cot \theta~$$

A differential right triangle of sides $( r,r',\sqrt{r^2+r^{'2}})$ can be drawn and it has trig relation $$\tan \psi=\frac{r}{r'}$$

Replacing $\psi$ by $ \pi/2-\psi$ for orthogonal trajectory with CCW sign convention... and with luck in separating variables in polar coordinates,

$$\frac{r'}{r}= \cot \theta,~~ \frac{dr}{r}=\cot \theta ~ d \theta ~;$$ Integrate with arbitrary constant $2A$

$$\log r = \log \sin \theta + \log 2A, ~ r= 2 A \sin \theta; $$

The given set are all circles through the origin centered on x-axis diameter $2C$ and the O.T. are all circles through the origin diameter $2A$ centered on y-axis.

It is recommended using another letter symbol for the O.T. set to visualize them as arbitrary, but different.