Final answer:
To solve the differential equation r''(t) = e^(12t-12), integrate the equation with respect to time: r''(t) = (1/12)e^(12t-12)+C1, r'(t) = (1/12)e^(12t-12)+C1*t+C2, r(t) = (1/144)e^(12t-12)+C1*t^2/2+C2*t+C3.
Step-by-step explanation:
The solution to the differential equation r''(t) = e^(12t-12) is found by integrating the second derivative twice with respect to time. To solve the differential equation r''(t) = e^(12t-12), we can integrate both sides with respect to t. Let's integrate the right side first.
The integral of e^(12t-12) with respect to t can be found using the power rule for integration, resulting in (1/12)e^(12t-12)+C1 (where C1 is the constant of integration). Now, we have the differential equation r''(t) = (1/12)e^(12t-12)+C1. To find r'(t), we integrate (1/12)e^(12t-12)+C1 with respect to t.
Which yields (1/12)e^(12t-12)+C1*t+C2 (where C2 is another constant of integration). Finally, we integrate (1/12)e^(12t-12)+C1*t+C2 with respect to t to find r(t), giving us (1/144)e^(12t-12)+C1*t^2/2+C2*t+C3 (where C3 is yet another constant of integration).