Fourth 3 week block: project details

For this 3 week block, use Laplace transforms to discuss the general solutions of the following differential equations. In each case plot solution curves for a variety of parameters and given initial conditions, and, for problems (2), (3) and (4), interpret the behavior of the solutions in terms of the physical problem.

(1) Some specific first and second-order liner differential equations:

(a) $\frac{dx}{dt}+x=e^{-t}$

(b) $\frac{dy}{dx}+y=cos(x)$

(b) $\frac{d^2y}{dx^2}+4\frac{dy}{dx}+4y=0$

Find the general solution using Laplace transforms (not by any other method) and plot solutions over a range of t values, for a given initial condition.

(2) Forced undamped motion: $\frac{d^2y}{dt^2}+\omega_0^2=Fsin(\omega t+\beta)$ .

Ordinary Differential Equations, Tenenbaum & Pollard, 338-342.

(3) Kirchoff’s law for the current in a closed circuit: $\frac{d^2i}{dt^2}+\frac{R}{L}\frac{di}{dt}+\frac{1}{CL}i=\frac{F\omega}{L}cos(\omega t+\beta)$ .