Final answer:
To solve the differential equation dy/dt = 2y - 3H(t - 1) with y(0) = 4, we integrate the equation to find y(t) = 4e^{2t} for t < 1. For t >= 1, the Heaviside step function affects the equation, and further integration is needed to find the complete solution.
Step-by-step explanation:
To solve the initial value problem dy/dt = 2y - 3H(t - 1), where H(t - 1) is the Heaviside step function and the initial condition is given as y(0) = 4, we need to integrate the differential equation. First, solve the homogeneous equation dy/dt = 2y, which is separable:
\( \frac{dy}{y} = 2 dt \)
Integrating both sides gives ln(y) = 2t + C, and taking the exponential of both sides gives the general solution y(t) = Ce^{2t}. To find the constant C, we use the initial condition:
y(0) = Ce^{0} = C = 4, so the solution before t = 1 is y(t) = 4e^{2t}.
After t = 1, we need to consider the Heaviside step function, which adds a piece-wise constant function. Thus:
y(t) = 4e^{2t} - 3 \int_1^t e^{2(t-s)} ds, for t \geq 1.
Evaluating the integral and coming up with a continuous solution at t = 1 will provide the complete solution. It is left as an exercise for the student to finish the solution and verify that it satisfies both the differential equation and the initial condition.