please help

1. (12 pts) Morphine is cleared from the body at a rate proportional to the amount present with a rate constant k given in units of hour1. A patient is given an injection of mini mg morphine immediately after surgery. The subsequent doses occur every 1- hours and are 0.51mi\" mg morphine. This continues indenitely. The patient has no morphine in her system before the rst injection. (a) (4 pts) Treating each injection as an instantaneous event, write down an Initial Value Problem to model the quantity of morphine m(t) in the patient's body as a function of time. (b) (3 pts) Solve the IVP. Express the solution in terms of the Heaviside functions u(t or) where n = U, 1, 2, . . . (c) (3 pts) Sketch the graph of the solution. You can use a graphing tool to get an idea of what it looks like but the sketch should be hand-drawn. (n\\ 1... . .1. . 1 1' . .. \\ no. 1 .. .1 n n