Final answer:
To solve the given differential equation: ((e^(2y)) - y)cos(x) * (dy/dx) = (eʸ)sin(2x), we can separate the variables and integrate.
Step-by-step explanation:
To solve the given differential equation: ((e^(2y)) - y)cos(x) * (dy/dx) = (eʸ)sin(2x), we can separate the variables and integrate. First, divide both sides by (e^y)cos(x) to get: (dy) / ((e^y)cos(x)) = sin(2x) dx. Next, integrate both sides with respect to their respective variables. The integral of dy / ((e^y)cos(x)) is ln|sec(x) + tan(x)| + C1, and the integral of sin(2x) dx is -cos(2x) / 2 + C2. Combining these results, we get ln|sec(x) + tan(x)| + C1 = -cos(2x) / 2 + C2. This is the general solution to the differential equation.