Final answer:
The solution involves separating variables of the differential equation, integrating both sides, and then applying the initial condition to find the specific solution for u(t).
Step-by-step explanation:
To solve the separable differential equation du/dt = e^3u + 8t with the initial condition u(0)=2, we'll separate variables and integrate both sides. First, rewrite the equation as du/(e3u) = dt, since the term 8t can be treated as a separate integration with respect to t. Integrate both sides to find the antiderivatives:
- For the left side: ∫ du/(e3u)
- For the right side: ∫ (8t dt)
After integrating, apply the initial condition to solve for the integration constant. This will give you the explicit equation for u(t).