Final answer:
To solve for r(t), we integrate r'(t) which is given in four options and apply the initial condition r(0) = < 4,5,6 >. Upon integration, we find the constants using initial conditions for three viable solutions, while the fourth, involving the natural log, is rejected due to being undefined at t=0.
Step-by-step explanation:
To find the solution r(t) of the differential equation with the given initial condition r'(t) and r(0) = < 4,5,6 >, we integrate each of the given possible derivatives:
- For a, integrate 4t + 5t^2 + 6t^3 to get the position function r(t).
- For b, integrate 4e^t + 5e^(2t) + 6e^(3t).
- For c, integrate 4cos(t) + 5sin(t) + 6t.
- For d, integrate 4ln(t) + 5√(t) + 6/t, but note that this would be undefined at t=0, so it cannot satisfy the initial condition.
After finding the antiderivatives, apply the initial condition to solve for the constants of integration. This process will yield the specific function r(t) for each case that satisfies both the differential equation and the initial conditions set forth.