Differential Equations

Find the differential equations of the following family of curves.

1. Parabolas with axis parallel to the y – axis with distance vertex to focus fixed as a.

2. Parabolas with axis parallel to the x – axis with distance vertex to focus fixed as a.

3. All ellipses with center at the origin and axes on the coordinate axes.

4. Family of cardioids.

5. Family of 3 – leaf roses.

1

(1) Upwards and downwards parabolas with latus rectum equal to 4a.

$y - k = \pm 4a(x - h)^2$

$y' = \pm 8a(x - h)$

$y'' = \pm 8a$

y−k=±4a

_{(x−h)}2 is this the equation for parabolas?isn't it 4a(y-k)=

_{(x-h)}2 ?I made a mistake in there, it should be (

x-h)^{2}= ±4a(y-k). The solution for (1) should go this way:$(x - h)^2 = \pm 4a(y - k)$

$2(x - h) = \pm 4ay'$

$2 = \pm 4ay''$

$y'' = \pm \frac{1}{2a}$

y−k=±4a

_{(x−h)}2 is this the equation for parabolas?isn't it 4a(y-k)=

_{(x-h)}2 ?Yes you are right, please refer to my reply above

so for number 2,

_{x-h=±4a(y-k)}2x'=±8a(y-k)

x''=±8a

am i wrong?

It is better to express your answer in terms of

y'rather thanx'. Althoughx'will do and simpler.$(y − k)^2 = \pm 4a(x − h)$

$2(y − k)y' = \pm 4a$

$y' = \dfrac{\pm 2a}{y - k}$

$y'' = \dfrac{\mp 2a \, y'}{(y - k)^2}$

$y'' = \dfrac{\mp 2a \, y'}{\left( \dfrac{y'}{\pm 2a} \right)^2}$

$y'' = \dfrac{\mp 8a}{y'}$

$y'' \, y' = \pm 8a$

our prof. gave the same question but the ans. he gave is different from your ans. in no 2, his answer is (y')^3 + 2ay"=0 are they the same with your answer sir ?

They are different. I made a mistake in $y - k$ of the above solution.

$(y − k)^2 = \pm 4a(x − h)$

$2(y - k) \, y' = \pm 4a$

$y' = \dfrac{\pm 2a}{y - k}$

$y - k = \dfrac{\pm 2a}{y'}$

$y'' = \dfrac{\mp 2a \, y'}{(y - k)^2}$

$y'' = \dfrac{\mp 2a \, y'}{\left( \dfrac{\pm 2a}{y'} \right)^2}$

$y'' = \dfrac{\mp 2a \, y'}{\dfrac{4a^2}{(y')^2}}$

$y'' = \dfrac{\mp (y')^3}{2a}$

$2a \, y'' = \mp (y')^3$

$\pm (y')^3 + 2a \, y'' = 0$

Thank You so much sir.

sir, do you know the standard or general equations for the last 3 problems?

what equations should i use?

I think you are from ADZU haha

Good evening, sir. Do you have the solution for the last three problems? Thank you!

Sir do you know how to eliminate arbitrary constant here in this equation?

(y-33)²=4a(x-h)?

## Add new comment