Solving the differential equation dy/dx = sin(xy)cos(xy)

To solve the given differential equation, we can use the method of separation of variables. This involves isolating the variables y and x on opposite sides of the equation and integrating both sides.

1. Separating the variables
We start by separating the variables on opposite sides of the equation:

dy/dx = sin(xy)cos(xy)

Next, we can rewrite the right-hand side using the double angle identity for sine:

dy/dx = (1/2)sin(2xy)

2. Integrating both sides
Now, we can integrate both sides of the equation with respect to x:

∫ dy = ∫ (1/2)sin(2xy) dx

The integral of dy with respect to y is simply y, and we can integrate the right-hand side using a u-substitution. Let's set u = 2xy, then du = 2xdy:

y = (1/2) ∫ sin(u) du

3. Evaluating the integral
Integrating sin(u) with respect to u gives us -cos(u):

y = (1/2)(-cos(u)) + C

Substituting back u = 2xy:

y = (1/2)(-cos(2xy)) + C

where C is the constant of integration.

4. Final solution
Therefore, the general solution to the given differential equation is:

y = (1/2)(-cos(2xy)) + C

where C can be any constant.

Summary:
To solve the differential equation dy/dx = sin(xy)cos(xy), we used the method of separation of variables. After separating the variables, we integrated both sides with respect to x. By evaluating the integral and substituting back, we obtained the general solution y = (1/2)(-cos(2xy)) + C, where C is the constant of integration.