Final answer:
To solve the differential equation, the process involves identifying knowns, separating variables, integrating both sides, applying initial conditions to find the integration constant, and then obtaining the explicit solution. Once the position function is found, velocity is determined by differentiating this function.
Step-by-step explanation:
To solve the differential equation and provide an explicit solution, the given information needs to be organized into knowns and unknowns. If, for example, the equation given is of the form dx/dt = f(t,x) where f(t,x) is a function of t and x, we would typically separate variables and integrate both sides to find an explicit function x(t).
Without the specific differential equation, here is a general step-by-step approach:
- Identify the knowns from the equation.
- Separate the variables if possible (move all x terms to one side and all t terms to the other).
- Integrate both sides of the equation, introducing a constant of integration C on the side where the variable of integration is not present.
- To find the value of the constant C, use any given initial conditions, such as x(0).
- Combine all terms to get the explicit solution x(t) in terms of t.
- Check the solution by differentiating and ensuring it satisfies the original differential equation.
If the problem includes an initial value such as x(0) = 0, then the constant C can be determined and the explicit solution would be a function x(t) that can be evaluated for any t. For example, if the integrated function was x(t) = (3/4)t^2 - t^3 + C and we apply the initial condition x(0) = 0, we find that C = 0 and our solution simplifies to x(t) = (3/4)t^2 - t^3.
The velocity can then be found by differentiating the position function x(t) to get v(t), where for example v(t) = 5t(1 - t), and evaluating it at the given times to ensure it matches the provided initial velocity conditions.