2. (7 pts) When an ODE needs to be solved repeatedly with different inhomogenous terms, it is often more efficient to calculate the Transfer Function and use convolution to get the solution for each inhomogeneous term. For example, suppose you have an LC circuit with a power source (g(t)) that can be set to deliver different voltage profiles, like a decaying voltage (g(t) = e-") or an AC source (g(t) = sin(wt)). (These are NOT the g(t) functions that we will use here - these are just examples.) L C g(t) Instead of solving y" + 4y' + by = g(t), y(0) =0, y'(0) =0 separately for each possible g(t), we can solve h" + 4h' + 5h =6(t), h(0) =0, h'(0) =0 and then use convolution with our now-known h(t), and any g(t), to solve for y(t): y(t ) = / g(t - =)h(z)dz Calculating h(t) and then this convolution integral for y(t) is generally more efficient than solving the ODE anew for each different g(t). Now, consider the given ODEs above for y(t) and h(t). (a) (3 pts) Calculate h(t) for this circuit using Laplace transforms. (b) (4 pts) For g(t) = eat, use h(t) and convolution to calculate the solution y(t). You should find yourself needing to calculate fo es sin(2)dz. Feel free to use WolframAlpha.com for this