Final answer:
The student's question involves solving an initial value problem for an ordinary differential equation with an initial condition. An integrating factor is used for the solution process, and the initial condition helps determine the integration constant for the exact solution.
Step-by-step explanation:
The question asks us to solve the initial value problem dx/dt − 5x = cos(2t) given that x(0) = 5. This is an ordinary differential equation (ODE) with an initial condition.
To solve this, we first need to find the integrating factor, which will be e^{-5t}, based on the coefficient of x in the ODE. Multiplying through by this integrating factor yields:
e^{-5t} dx/dt - 5e^{-5t}x = e^{-5t} cos(2t).
The left side of this equation is the derivative of e^{-5t}x. Thus, we integrate the right side with respect to t, which gives us:
∫ e^{-5t} cos(2t) dt.
After integrating, we solve for the constant of integration using the initial condition x(0) = 5. With the solution x(t), the function's velocity could also be derived, which is the first derivative of the position function with respect to time.
However, note that x(0) = 5 is different from the zero initial condition mentioned in the provided information. Therefore, the latter part of the information concerning solving for velocity using the same initial condition as x(0) = 5 doesn't apply directly to this problem. This complication should be considered while computing the integration constant.