hey , i got 6 matlab problems in differential equations

MAT 275 Laboratory 2

Solving Spring/Mass Systems

Consider the following differential equation form:

𝑥′′ + 2𝑥′ + 𝜔2𝑥 = 𝐹(𝑡)

Use springmass.m to generate a picture of motion for each question below (1-3). Provide the

final solution y(t) for each problem below, using the general solution statement for questions 1, 2

and 3.

(1) Under-damped harmonic oscillator 𝑥′′ + 0.2𝑥′ + 2𝑥 = 0, 𝑥(0) = 1, 𝑥′(0) = 2, 𝑡𝑓 = 12𝜋.

Hints:  = 0.1 & 𝜔 = √2. For  < 𝜔,

𝑥(𝑡) = 𝐴𝑒−𝑡 cos (√𝜔2 − 2 𝑡 − )

(2) Critically damped harmonic oscillator 𝑥′′ + 2𝑥′ + 𝑥 = 0, 𝑥(0) = 1, 𝑥′(0) = 2, 𝑡𝑓 = 12𝜋.

Hint: For  = 𝜔, 𝑥(𝑡) = 𝐴𝑒−𝑡 + 𝐵𝑡𝑒−𝑡

(3) Over-damped harmonic oscillator 𝑥′′ + 4𝑥′ + 𝑥 = 0, 𝑥(0) = 1, 𝑥′(0) = 2, 𝑡𝑓 = 12𝜋.

Hint: For  > 𝜔,

𝑥(𝑡) = 𝐴𝑒𝑟+𝑡 + 𝐵𝑒𝑟−𝑡 , 𝑟 = − ± √2 − 𝜔2

For questions 4 , 5, and 6, use springmassdriven.m to generate a picture of motion. For each

question below provide a picture as stated above, the final solution y(t) for each problem and

provide a breakdown of the transient and steady-state solutions with a description of motion.

(4) Driven undamped harmonic oscillator 𝑥′′ + 𝑥 = cos (2𝑡), 𝑥(0) = 1, 𝑥′(0) = 0, 𝑡𝑓 = 24𝜋.

(5) Resonantly driven undamped harmonic oscillator 𝑥′′ + 𝑥 = cos (𝑡), 𝑥(0) = 1, 𝑥′(0) = 0, 𝑡𝑓 = 24𝜋.

(6) Resonantly driven damped harmonic oscillator 𝑥′′ + 0.2𝑥′ + 2𝑥 = cos(𝑡) , 𝑥(0) = 1, 𝑥′(0) = 0, 𝑡𝑓 = 24𝜋.