Solve the following differential equations where the input

Question

Systems Modeling Laboratory

Problem 1- Solve the following differential equations:

(i) dy/dt + 2y =12              y(0) = 10

(ii) d²y/dt² + 3(dy/dt) + 2y = 24                   y(0) = 10  y'(0) = 0           

(iii) d²y/dt² + 2(dy/dt) + 5y = 20                 y(0) 0       y'(0) = 10                          

Problem 2- Solve the following differential equations where the input is f(t) = 5t and the initial conditions are zero. Plot the response of the following models for 0 ≤ t ≤ 1.5

(i)  3x^·· + 21x^· + 30x = f(t)                      

(ii) 5x^·· + 20x^· + 65x = f(t)                      

(iii) 4x^·· + 32x^· + 30x = 3f^·(t) + 2f(t)                      

Problem 3 - Obtain the Laplace transform of the following function:

x(t) = t

x(t) = (2 - ae^-2t + be^-3t )

x(t) = 10cos(2t)

Problem 4 - Calculate the Laplace transform of the following function:

v(t) = 3e^-2t sin 5t + 4e^-2t cos5t

Problem 5 - Calculate the inverse transform of the following functions:

(i) F(s) = 100(s + 3)/(s+1)(s+2)(s²+2s+5)

(i) Y(s) = (10/(s + 2)) + (48/(s+2)(s²+16))

(iii) Expand the following transform using Partial-Fraction expansion. Also calculate the inverse transform.

X(s) = 4s + 3/s(s² + 6s + 34)

Problem 6- Obtain the inverse Laplace transform x(t) for each of the following transforms:

(i) X(s) = 7s+2/s²+6s+34

(ii) X(s) = 4s+3/s(s²+6s+34)

Problem 7 - Solve the following problems:

(i) x^·· + 7x^· + 10x = 20    x(0) = 5     x^·(0) = 3

(ii) 5x^·· + 20x^· + 20x = 28    x(0) = 5     x^·(0) = 8