Can I get the solution for these physics and Mathematical problems?

textbooks you might need:

? Alec J. Schramm, "Mathematical Methods and Physical Insights"

? A. Altland, J. von Delft, "Mathematics for Physicists: Introductory Concepts and Methods"

? Mary L. Boas, "Mathematical Methods in the Physical Sciences"

Exercise 1. Linear first order ODE A population of bacteria is exposed to a toxin. The corresponding death rate is proportional to the number n(t) of the living bacteria and to the amount of toxin T(t) at time t. Altogether the death rate equals Tn(t)T(t), where T is a positive constant. On the other hand, bacteria reproduce exponentially with a rate of yn(t) for y > 0. This yields the following differential equation for the number of bacteria: n(t) = yn(t) - in(t) I(t). Note that this is a linear first order ODE for the dependent variable n(t), but it does not have constant coefficients because T(t) depends on t. a) Assume that the toxin is being supplied at the constant rate T(t) = at starting from t = 0, where a > 0. Use separation of variables to solve the differential equation with the initial condition n(t = 0) = no. b Show that the population of bacteria grows until the time t = var and then diminishes. Show furthermore that for t - co the bacteria are annihilated, i.e. n(t) - 0. Make a sketch of n(t). c) How long does it take to reduce the population to half its original size? Exercise 2. Harmonic oscillator In the lecture (see also Example 5.6 in the graduate lecture notes) we solved the homogeneous harmonic oscillator ODE for the RLC circuit, Q " (t ) + I PQ' (t ) + ICQ (t ) = 0, (1) with the ansatz Q(t) = emt, resulting in the equation R 1 = 0= m+ R R2 4 L2 - LC We denote the term inside the square root as D= R2/L2 - 4/LC. Depending on the value of D, the oscillator is either overdamped (D > 0), underdamped (D