Final answer:
To solve the initial value problem y''-y=t², we can find the general solution to the homogeneous equation and then the particular solution to the non-homogeneous equation. By using the initial conditions, we can determine the values of the constants in the solutions. The solution for velocity is obtained by differentiating the solution for position.
Step-by-step explanation:
Solution for Position
We can solve the initial value problem by finding the general solution to the homogeneous equation y'' - y = 0 and then finding the particular solution to the non-homogeneous equation y'' - y = t^2. The general solution to the homogeneous equation is y = c1e^t + c2e^-t, where c1 and c2 are constants. To find the particular solution, we can use the method of undetermined coefficients and guess a particular solution of the form yp = At^2 + Bt + C, where A, B, and C are constants. Substituting this particular solution into the non-homogeneous equation, we find that yp'' - yp = t^2. By equating coefficients, we can solve for A, B, and C. Once we have the particular solution, we can add it to the general solution to get the complete solution to the non-homogeneous equation. Finally, we can use the initial conditions y(0) = 2 and y'(0) = 3 to find the values of the constants c1 and c2.
Solution for Velocity
To find the solution for velocity, we can differentiate the solution for position with respect to time. This will give us the expression for velocity, which is y' = cy1'e^t + c2'e^-t + A2t + B, where c1' and c2' are constants. Substituting y(0) = 2 and y'(0) = 3 into this expression, we can solve for the values of c1' and c2'. The complete solution for velocity is then y' = cy1'e^t + c2'e^-t + A2t + B.