What is Clairaut's Equation and How Does it Apply to Optics?

[jnbfive]

Damn, alright. Even though I know that's right, you're saying to separate the variables to get the equation to say dy = e^x * dx and then integrate.

 

[maze]

No, you had it right the first time.* (d/dx)e^x - e^x = e^x - e^x = 0.

this isn't a trick question or anything

Now the next question:
If f = y*x^2, and x = y+1, then what is f in terms of y only?

*doing the integral thing will work too though, but that's much harder if you already have a solution and you just want to test it.

 

[jnbfive]

f= y^3 +2y^2 +y

 

[maze]

Yeah sure.

So now, if x = sqrt((y+K)^2 - y^2), then verify that it solves the DE (dx/dy) - (-y+(x^2+y^2)^(1/2))/x = 0

 

[jnbfive]

dx/dy = k/((2ky+y^2)^(1/2))

k/((2ky+y^2)^(1/2)) - (-y+((sqrt((y+K)^2 - y^2))^2 +y^2)/sqrt((y+K)^2 - y^2)

k/((2ky+y^2)^(1/2)) - (-y+(y+K)^2+y^2-y^2)^(1/2) / sqrt((y+K)^2 - y^2)

k/((2ky+y^2)^(1/2)) - (-y+((y+K)^2+y^2-y^2)^(1/2) / sqrt((y+K)^2 - y^2)

k/((2ky+y^2)^(1/2)) - (-y+(y+K)) / sqrt((y+K)^2 - y^2)

k/((2ky+y^2)^(1/2)) - k/((2ky+y^2)^(1/2)) = 0

 

[maze]

Big pimpin.

So basically you solved the equation using the strategies in the page, and now you know your answer is correct because you plugged it into the original DE. Sounds good to me.

 

[jnbfive]

So for part L, how would I find y = f(x). Same strategies?

 

[maze]

Ok, let's go simple to complex again
If x=y/2+1, then what is y in terms of x?

 

[jnbfive]

y= 2x-2

 

[maze]

Yeah definitely.
Now, if x^2 = (y+k)^2 - y^2, what is y in terms of x?

 

[jnbfive]

y = (x^2+K^2)/(2K)

 

[maze]

And what sort of curve is it?

Like, is it a straight line? Is it an exponential? Is it a sine wave? An ellipse? etc.

 

[jnbfive]

Looks like a parabola, but I also want to say I've seen something else like this such as cosine.

 

[maze]

Yep, its just a parabola.

Such a simple result for such a complicated looking DE.

Does this all make sense then, or are there still some confusing bits?

 

[jnbfive]

No, this makes sense. I actually came up with this stuff like an hour or two ago, but like you said, it looked so simple, I thought that there was no way that it could be right and that I was missing a crucial step and making dumb mistakes like I usually do. Thank you so much for your help. You probably just helped me pass my differential equations class!

 

[maze]

Have more confidence in your math skills!

 

[flexflow]

I think you didn't get the answer of w(y)

if you substitute well, you will get (dw/dy)^2+...

That is not second order ODE, just square of (dw/dy)

I think your answer is from second order or something like that.