Final answer:
To solve the given equation (3e^x sin(y) - 3y)dx - (3x 3e^x cos(y))dy = 0, we can use the method of exact equations. By finding an integrating factor and integrating both sides, we obtain e^xy(3sin(y) - 3y) = C.
Step-by-step explanation:
To solve the given equation:
(3exsin(y) - 3y)dx - (3x 3excos(y))dy = 0
We can see that this is a linear first-order partial differential equation (PDE) of the form M(x,y)dx + N(x,y)dy = 0, where M(x,y) = (3exsin(y) - 3y) and N(x,y) = - (3x 3excos(y)).
We can use the method of exact equations to solve this PDE by finding a integrating factor. The integrating factor can be found by dividing the coefficient of dx by M(x,y) and dividing the coefficient of dy by N(x,y). In this case, the integrating factor is exy. Multiplying the entire equation by exy, we get:
(3e2xsin(y) - 3yexy)dx + (-3x 3e2xcos(y)exy)dy = 0
This can be rewritten as:
d(exy(3sin(y) - 3y)) = 0
Integrating both sides, we get:
exy(3sin(y) - 3y) = C
where C is a constant.