Separable Differential Equation dy/dx

[aznkid310]

Homework Statement

dy/dx =[cos^2(x)][cos^2(y)]

Homework Equations

The solution to this problem is y = +/- [(2n + 1)*pi]/4

How? Do i just plug C back into the equation? That seems a little messy

The Attempt at a Solution

dy/cos^2(y) = cos^2(x) dx

After integrating: (1/2)tan(2y) = (1/2)(x + cos(x) + sin(x) + C)

C = 2tan(2y) - 2x -sin(2x) for cos(2y) not = 0

 

[DavidWhitbeck]

You have not supplied enough information to make your solution unique. What point must the curve pass through?

 

[tiny-tim]

dy/cos^2(y) = cos^2(x) dx

After integrating: (1/2)tan(2y) = (1/2)(x + cos(x) + sin(x) + C)

C = 2tan(2y) - 2x -sin(2x) for cos(2y) not = 0

Hi aznkid310!

No … you're getting your ^2 and your 2 mixed up … it's just tany on the left.

And on the right … you've gone all weird!

Use cos²x = 1/2(1 + cos(2x)).

 

[aznkid310]

It just saved solve the differential equation. But the solution included all of that.

As for the integration, i checked w/ an integral calculator and the left side is correct. For the right side, its actually cos^2(2x), my mistake.

 

[D H]

The solution to this problem is y = +/- [(2n + 1)*pi]/4

Firstly, its $y=\frac{2n+1}2\pi$. You don't need the $\pm$ and the denominator is 2, not 4.

Secondly, this is not "the" solution. There are infinitely many solutions you can find via integrating the differential equation. Hint: You will not get these particular solutions by integrating the differential equation. Big hint: what is the derivative of $y=c$ with respect to $x$?

Thirdly, your integration ran afoul somewhere.