Final answer:
To solve the differential equation y' = 4xy - 2x with initial condition y(0)=8, we use an integrating factor e2x2, integrate both sides, apply the initial condition, and solve for y(x).
Step-by-step explanation:
The problem presented is a first-order differential equation that can be solved using an integrating factor. The given differential equation is y' = 4xy - 2x, and we are given the initial condition y(0)=8. To solve for y(x), we first find an integrating factor that will allow us to convert the left-hand side of the differential equation into the derivative of a product. An integrating factor for this equation is e∫4x dx = e2x2. Multiplying both sides of the differential equation by this integrating factor allows us to write the left-hand side as the derivative of y times the integrating factor.
After integrating both sides and applying the initial condition y(0)=8, we can solve for the function y(x). This process involves integrating, applying the initial condition, and algebraically manipulating the expression to isolate y(x).