Final answer:
To solve the given differential equation (3eˣsiny-3y)dx (-3x 3eˣcosy)dy=0, start by separating the variables, then integrate and solve for y. The solution is y = (2eˣsiny + C)/(3x).
Step-by-step explanation:
To solve the given differential equation: (3eˣsiny-3y)dx - (-3x 3eˣcosy)dy = 0, we can start by separating the variables and then integrating.
Step 1: Separate the variables:
(3eˣsiny - 3y)dx = (-3x 3eˣcosy)dy
Step 2: Integrate both sides:
∫(3eˣsiny - 3y)dx = ∫(-3x 3eˣcosy)dy
Step 3: Simplify and solve the integrals:
3eˣsiny - 3xy = -3eˣsiny + C
Step 4: Solve for y:
3xy = 2eˣsiny + C
Therefore, the solution to the differential equation is y = (2eˣsiny + C)/(3x).