Mathematics 1220 PRACTICE EXAM IV Spring 2002

1. Solve the following differential equations.

(a) xy′ + (1 + x)y = e^{−x}, y(1) = 0 (b) y′′ + 5y′ − 6y = 0

(c) y′′ − 4y′ + 4y = 0 (d) y′′ + 4y = 0, y(0) = 0 and y′(0) = 1

(e) y′′ + y′ + y = 0 (f) y′′ + y′ − 6y = 2x^{2} (g) y′′ + 4y′ = cos x

2. Consider a microwave propagating vertically with electric field in the x−direction at time t given by E(x, t). Assuming the wave is time harmonic with one angular frequency ω, then E(x, t) = e^{iωt}E(x). If the wave propagates through a medium (such as a turkey) described by the permittivity, or dielectric constant, ϵ(x) (related to the index of refraction n(x) by n = \sqrt{ϵ}), then E(x) satisfies the second order non-

constant coefficient differential equation \frac{d^{2}E}{dx^{2}} + k_{0}^{2}ϵ(x)E(x) = 0, where k_{0} = ω/c is the free space wave number, and c is the speed of light. For a homogeneous medium, such as free space where ϵ(x) = 1, E satisfies

\frac{d^{2}E}{dx^{2}} + k_{0}^{2}E = 0.  (1)

Use the characteristic polynomial of (1) to find its general solution, and then from

your general solution find the particular solution satisfying E(0) = 0 and \frac{dE}{dx} (0) = 1.

3. Consider a pendulum of length ℓ. The angle θ(t) that the pendulum makes with respect to vertical at time t satisfies the second order nonlinear differential equation

ℓ\frac{d^{2}θ}{dt^{2}} = − g sin θ, where g is the acceleration due to gravity. For small oscillations (small θ), sin θ can be approximated by θ, which yields the linearized equation

\frac{d^{2}θ}{dt^{2}} + ω^{2}θ = 0, ω = \sqrt{\frac{g}{ℓ}} .  (2)

Use the characteristic polynomial of (2) to find its general solution, and then from your general solution find the particular solution with initial angle θ(0) = π/16 and

initial angular velocity \frac{dθ}{dt} (0) = 0.

4. Know how to solve the logistic equation.