Answer:
To find the particular solution of the differential equation y' = e^2 - 2x^3 with initial condition y(0) = 4, we need to integrate both sides of the equation with respect to x and then solve for the constant of integration using the initial condition.
Integrating both sides, we get:
y(x) = ∫(e^2 - 2x^3) dx
y(x) = e^2x - (1/2)x^4 + C
where C is an arbitrary constant of integration.
To find the value of C, we use the initial condition y(0) = 4:
4 = e^2(0) - (1/2)(0)^4 + C
4 = 1 + C
C = 3
Therefore, the particular solution of the differential equation y' = e^2 - 2x^3 with initial condition y(0) = 4 is:
y(x) = e^2x - (1/2)x^4 + 3