Final answer:
To solve the given differential equation, use the method of integrating factors and rearrange the equation to standard form. Identify the integrating factor, multiply both sides of the equation by the integrating factor, integrate, and solve for y to obtain the solution.
Step-by-step explanation:
To solve the given differential equation x(dy/dx)-2y=x³cos(4x), we can use the method of integrating factors. First, we rearrange the equation to the standard form of a linear equation: dy/dx - (2/x)y = x²cos(4x). Then, we identify the integrating factor e^(∫(-2/x)dx) = e^(-2ln(x)) = x^-2. Next, we multiply both sides of the equation by the integrating factor and integrate. Finally, we solve for y to get the solution of the differential equation.