Separable Differential Equation dy/dx

In summary: I don't know where and won't bother to try to find where, since it is irrelevant to the differential equation. The best way to get the other solutions is to graph the curve (and the solutions I have been talking about are not on the curve).In summary, the solution to the given differential equation is y = (2n+1)pi/2 where n is any integer. However, there are infinitely many other solutions that can be found by integrating the equation, but they are not on the curve of the given solution.
  • #1
aznkid310
109
1

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
 
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  • #2
You have not supplied enough information to make your solution unique. What point must the curve pass through?
 
  • #3
aznkid310 said:
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! :smile:

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! :rolleyes:

Use cos²x = 1/2(1 + cos(2x)).
 
  • #4
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.
 
  • #5
aznkid310 said:
The solution to this problem is y = +/- [(2n + 1)*pi]/4
Firstly, its [itex]y=\frac{2n+1}2\pi[/itex]. You don't need the [itex]\pm[/itex] 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 [itex]y=c[/itex] with respect to [itex]x[/itex]?

Thirdly, your integration ran afoul somewhere.
 
Last edited:

1. What is a separable differential equation?

A separable differential equation is a type of differential equation where the dependent variable and independent variable can be separated and rearranged on opposite sides of the equation. This allows for the equation to be solved using integration.

2. How do you solve a separable differential equation?

To solve a separable differential equation, you must first separate the dependent and independent variables on opposite sides of the equation. Then, you can integrate both sides with respect to their respective variables and add a constant of integration to the final solution.

3. What is the general form of a separable differential equation?

The general form of a separable differential equation is dy/dx = f(x)g(y), where f(x) and g(y) are functions of x and y, respectively.

4. Can a separable differential equation have more than one solution?

Yes, a separable differential equation can have infinite solutions due to the constant of integration that is added during the integration process. This constant can take on any value, thus resulting in multiple possible solutions.

5. What are some real-life applications of separable differential equations?

Separable differential equations are commonly used in physics, chemistry, and engineering to model a variety of natural phenomena such as population growth, radioactive decay, and chemical reactions. They are also used in finance to model interest rates and in economics to model supply and demand.

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