Question

Part A is right i just need help with B

Consider a radioactive material with a decay constant of 2 per month, and every month we add 3 grams of material. Then, the radioactive material is described by the differential equation 2y3. (a) Find an explicit expression of all solutions y of the differential equation above. Denote by c any arbitrary integration constant. vlt) (exp(-2t-c)-3)/2 (b) Suppose the initial amount of material is 4 grams. From all the solutions above, find the only solution that satisfies the initial condition v(0) 4 y(t) -(exp(-2t+In(5))-3)/2

Answer 1

y' = -2y + 3 dy/dx = -2y + 3 dy/-2y + 3 = dx dy/-2y + 3 = dx dy-2 (y - 3/2) dx dy/(y - 3/2) = -2 dx take integration integral dy/y - 3/2 = -2 integral dx in (y - 3/2) = -2x + C y - 3/2 = e^(-2x + c) y = e^-2x + 6 + 3/2 y = e^-2x e^c + 3/2 take e^c = c_1 y = e6-2x c_1 + 3/2 \r\n here y (0) = 4 4 = e^-2 (0) C_1 + 3/2 4 = e^-0 c_1 + 3/2 4 = (1) c_1 + 3/2 4 = c_1 + 3/2 c_1 = 4 - 3/2 c_1 = 8 - 3/2 c-1 = 5/2 put it back in eqn y = e^-2x 5/2 + 3/2 y = 5/2 e^-2x + 3/2