An interesting math problem!

[Guest SNJERIN]

While I was starting solving differential equations, I ran into an interesting problem! So i thought it might be a good idea to share it with you guys who will take the calculus option or already taking it.   

Edited August 14, 2014 by Haitham Wahid

[IctEcon]

do you have the actual solution?

[jwmurri]

I don't know what you want to know about this problem, but k=3, in case you were wondering. You have to separate the expression on the right, integrate both sides, find the constant of integration, and then simplify it. 

[Ossih]

I think it's like this:

 

by separation, we get, dy/ √1-y^2 = dx/ √1-x^2,

Multiply both sides by -1 so -dy/ √1-y^2 = -dx/√1-x^2

integrating both sides, arccos y = arccos x + C

y = cos(arccos x + C)

y = cos(arccos x) cos C - sin(arccos x)sin C

y = x cos C - √1-x^2 sin C

=> 2y = 2x cos C - 2√1-x^2 sin C

 

now, sub in y = √3/2 and x= 1/2

 

√3 = 1 cos C - √3 sin C

√3/2 = 1/2 cos C - √3/2 sin C

√3/2 = cos(π/3 + C)

π/6 = π/3 + C

 

C = -Ï€/6

 

So, 2y = 2x * √3/2 - 2 * √1-x^2 * -1/2

so, 2y = x√3 + √1-x^2

 

so k = 3

 

P.S. I just realized I used arccos instead of arcsin, but it shouldn't matter cause I can put a negative on both sides and then if I integrate, its arccos. Just for clarity, I'm adding the minuses up there.

Edited August 15, 2014 by Ossih

[Guest SNJERIN]

Nice. I guess my way was a little bit different 

[IctEcon]

mine seemed to be correct aswell then I got k=3

Thankyou very much   was a good problem to get back into math-thinking before school starts

[Ossih]

Yeah, yours is basically the same as mine but with arcsin, and you use a substitution to make the simplification cleaner

[Emmi]

It was interesting to see you using arccos instead of arcsin, which is what I used to solve this problem as well. But with the negative signs it doesn't matter, because you'll end up with the same result anyways, just using a different trig identity to get the final answer.

 

My method of solving this was very similar to Ossih's way, only I used arcsin and a very very slightly different method of finding C:

 

[Ossih]

Yeah :$ Thing was I started using arccos accidentally and I finished and got the right solution and posted. Then I realised it was supposed to be arcsin and it was a mistake, but one that made no difference. But so that no one else gets confused, I edited and multiplied both sides by -1.

I feel so good at the moment cause we haven't even done the calculus option yet. It's a good touch up with trig and calculus.