1) If you have an inhomogeneous, linear equation like f''(x) + f'(x) + f = x you can find the general solution by finding a particular solution (with no constants of integration)to this equation and adding it to the general solution to the homogeneous equation

f''(x) + f'(x) + f = 0 (the complementary solution). Would this same technique work with a nonlinear equation such as f''(x) + ( f'(x))^2 + f = x ? In other words if you added a particular solution to a complementary solution for this equation, would it solve the inhomogeneous equation? You must justify your answer.

2) A more realistic driving force than the exponential would be a sinusoidal driving force. For example, a common
circuit tends to have resistors, capacitors, and inductors. (You don’t need to know what any of those are for this question. Because we don’t want to get into the physics of circuits we'll just going to leave all units off of this problem and assume
all numbers are given in SI units.) As noted in the book the equation for the
current in such a circuit is:

L( d^2 I/dt^2)+ R(dI/dt) + (1/C)(I) = 0

If you hook that circuit up to an alternating voltage source, such as a wall outlet, the equation becomes:

L(d^2I/dt^2) + R(dI/dt) + (1/C)I = A*sin(ù' t)

Consider a circuit with L = 1, R = 7, C = 1/12, A = 10, ù'= 120ð. Find the complementary and particular solutions for this driven circuit. (Hint: Search for a particular solution of the form
k1*sin(ù't)+ k2*cos(ù't).) Note this way mayb be harder, but you must solve this without complex numbers.

3) Solve f '(t) = ln( f ) + f using Euler’s method. f(0) = 10 and try to find f(2). Start with step size of 2, then 1, then 0.5, and continue to decrease your step size by a half until you find that doing so changes the result by less than 1%.

How did your answer change as you decreased the step size (increasing, decreasing, oscillating)? Explain why in words. If you had kept decreasing the step size further would you have expected this same behavior to continue indefinitely? Why or why not?

4) A damping force proportional to speed v is accurate for a slowly moving object such as a ball dropped in water. For a rapidly moving object such as a skydiver in free fall, the damping force is roughly proportional to v^2.
Consider a 1kg object dropped from rest at 100 m above the ground. If it experiences a damping force equal to v^2 (in SI units), how long will it take to reach the ground?

(Hint: Consider using Mathematica to help you with this.)