Math 1

In this assignment, you will be asked to watch a short video and perform some applications that have to do with solving second order differential equations with constant coefficients. These differential equations can be solved using the method of annihilators. There are two primary applications that we will be studying:

1. Spring Mass Systems 2. RLC Circuits

You can take any notes that you feel are necessary for you to understand the applications and solve for their models. What we are primarily looking for is the function that will model the particular situation based upon the differential equation. And then we will be discussing how we can use the D.E. and the model to ascertain key characteristics of the modeled situation.

Spring Mass Systems:

1. Basic setup of the spring mass system. Construct a diagram of the basic spring mass system setup and construct the differential equation. Make sure that you include what the variables in the differential equation mean.

2. Consider the following initial-value problem for the spring mass system.

𝑑𝑑2𝑦𝑦 𝑑𝑑𝑑𝑑2

+ 2 𝑑𝑑𝑦𝑦 𝑑𝑑𝑑𝑑

+ 𝑦𝑦 = 0, 𝑦𝑦(0) = −1, 𝑑𝑑𝑦𝑦 𝑑𝑑𝑑𝑑

(0) = 2

a. Is the system overdamped, underdamped or critically damped? Explain how you know. b. Determine if the system passes through the equilibrium position. If it does, give the

time t.

3. Suppose that you have a spring mass system. The mass on the system is 1 g. The spring mass constant is 25. In this situation there is no damping, but there is an external force acting on the system that is modeled by the function, 𝐹𝐹(𝑑𝑑) = 75 cos 5𝑑𝑑. Construct the differential equation for the spring mass system and then solve for 𝑦𝑦(𝑑𝑑).

a. In this situation, is there resonance? Explain how you know. b. Suppose that you let the spring mass continue to oscillate for a very long time, give a

function that models the movement after this very long time t (e.g. find the steady state of the system).

RLC Circuits 1. Construct a diagram of the basic RLC Circuit and construct the differential equation. Make sure

that you include what the variables in the differential equation mean. 2. Suppose that you have a circuit with a resistance of 2 Ω, inductance of 2

11 H and a capacitance of

11 60

F. An EMF with equation of 𝐸𝐸(𝑑𝑑) = 6 cos 4𝑑𝑑 supplies a continuous charge to the circuit. Suppose that the q(0)= 8 V and the q’(0)=7.

3. Determine the steady state solution and transient part of the circuits current.