Final answer:
To solve the initial-value problem dy/dt = e^5t, integrate both sides with respect to t, substitute the initial condition y₀ = 8, and solve for the constant of integration.
Step-by-step explanation:
To solve the initial-value problem dy/dt = e^5t, we can integrate both sides with respect to t. Since the derivative of e^5t is e^5t itself, the equation becomes:
∫dy = ∫e^5t dt
Integrating both sides gives y = (1/5)e^5t + C, where C is the constant of integration. To find the specific solution, we can use the initial condition y₀ = 8. Substituting this value into the equation:
8 = (1/5)e^5(0) + C
8 = (1/5) + C
C = 8 - 1/5 = 39/5
Therefore, the solution to the initial-value problem is y = (1/5)e^5t + 39/5.