Homework Help: Diff Equation problem- give me a nudge, please

1. Jun 26, 2011

Latecomer

1. The problem statement, all variables and given/known data

(dy/dx) - cos2(x-y) = 0

2. Relevant equations

3. The attempt at a solution

I'm unsure where to start this. This isn't a liner equation of form (dy/dt) + p(t)y = g(t) ; and it's not separable nor an exact equation.

Is it illegal to make this (dy/dx) - cos2(x) + cos2y = 0 ???

2. Jun 26, 2011

ehild

It is totally illegal!!:uhh:

Hint: replace y-x with a new variable.

ehild

3. Jun 26, 2011

Latecomer

Yeah, I knew that was illegal. I was just grasping at straws because I had no idea where to begin. And it's early...

Well, I have not worked with problems like this at all and I'm having trouble finding anything similar in my text, but this is what I have figured out so far:

(dy/dx) -cos2(x-y) = 0

(dy/dx) = cos2(x-y) make (x-y) = v

(dy/dx) = cos2(v)

v = x - y
y = x - v
y' = 1 - v'

from original : y' = cos2(x - y)
so: 1 - v' = cos2(v)
and : v' = 1 - cos2(v)

Am I heading in the right direction? Another nudge? Thanks again.

4. Jun 26, 2011

ehild

It is all right so far. Is not it a separable de?

ehild

5. Jun 26, 2011

Char. Limit

Remember that 1-cos2(v) = sin2(v).

6. Jun 26, 2011

Latecomer

Yes, I should have seen that.

(dv/dx) = 1-cos2(v)

dv/1-cos2(v) = dx

which is : csc^2(v) = dx

integrate:

-cot(v) = x + c

-cot (x - y) = x + c

Thank you for your help. I was just staring dumbly at it.

This implicit form should be a suitable answer for this question, yes? I was just told to solve the equation.

Last edited: Jun 26, 2011
7. Jun 26, 2011

Char. Limit

It should be suitable, but it's a simple process to get the answer in explicit form, if you want it.

8. Jun 26, 2011

Latecomer

Hmm....

-cot (x-y) = x + c

cot (x-y) = -x +c

(x - y) = arccot (-x + c)

-y= arccot (-x + c) - x

y = x - arccot (-x + c) ???

Last edited: Jun 26, 2011
9. Jun 26, 2011

Char. Limit

Hint: The inverse of the cotangent function is arccotangent, not arctangent.

10. Jun 26, 2011

Latecomer

hehe, yeah I saw that as soon as I submitted it and then edited it. You're fast

11. Jun 26, 2011

Char. Limit

Looks good to me. You can plug it in and test it if you want.