Exact Differential Equations for Level Curves of u(x,y) = cos(x^2-y^2)

In summary, the differential equation du = 0 is a cosine function with two roots (x=-2 and x=2). The level curves u(x,y) = constant are the solutions to this equation.
  • #1
Mark Brewer
38
4

Homework Statement



Given u(x.y), find the exact differential equation du = 0. What sort of curves are the solution curves u(x,y) = constant? (These are called the level curves of u).

u = cos(x2 - y2)

The Attempt at a Solution


partial derivative du/dx = (-2x)sin(x2 - y2)dx
partial derivative du/dy = (2y)sin (x2 - y2)dy

P = (-2x)sin(x2 - y2)dx ; Q = (2y)sin (x2 - y2)dy

I found that they're not exact differential equations

so, I'm using integration factors.

1/Q(P dx - Q dy) = R(x)

(1/(2y)sin (x2 - y2))((-2x)sin(x2 - y2) - (2y)sin (x2 - y2)) = R(x)

I get, R(x) = -x/y

Then,

F = exp^(R(x))dx

I get, F = exp^((-x^2)/(2y))

Then,

M = FP and N = FQ

so,

M = (exp^((-x^2)/(2y)))((-2x)sin(x2 - y2))

and

N = (exp^((-x^2)/(2y)))((2y)sin (x2 - y2))

I then took the partial derivative of M in respect to y, and this is where I am getting stuck.

I also have to take the partial derivative of N in respect to x.

Any help would be much appreciated.
 
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  • #2
At constant u,
$$du=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy=0$$
What does this tell you about dy/dx?

Chet
 
  • #3
Chestermiller said:
At constant u,
$$du=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy=0$$
What does this tell you about dy/dx?

Chet

I believe it tells me that the partial derivatives are continuous
 
  • #4
Mark Brewer said:
I believe it tells me that the partial derivatives are continuous

it may also tell me that the partials in respect to x and y are equal to u and 0...?
 
  • #5
Checking for exactness,
P = (-2x)sin(x2 - y2)dx, Q = (2y)sin (x2 - y2)dy

dP/dy = (4xy)cos(x^2- y^2)
dQ/dx = (4xy)cos(x^2 - y^2)

dP/dy = dQ/dx, they are exact!
So, don't use the "integration factors", just find the general solution.
 

1. What are differential equation curves?

Differential equation curves are mathematical functions that describe the relationship between a variable and its derivatives. They are commonly used to model real-world phenomena in fields such as physics, engineering, and biology.

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