One of the uses of the Laplace trans-form is the solution of differential equations.

(a) Suppose you are given the ordinary differential equation that rep-resents a LTI system,

y⁽²⁾(t) + 0.5y⁽¹⁾ (t) + 0.15y^(t) = x(t), t ≥ 0

where y(t) is the output and x(t) is the input of the system, y⁽¹⁾(t) and y⁽²⁾(t) are first- and second-order derivatives with respect to t. The input is causal, i.e., x(t) = 0,t < 0. What should the initial conditions be for the system to be LTI? Find Y(s)for those initial conditions.

(b) If y⁽¹⁾(0) = 1 and y(0) = 1 are the initial conditions for the above ordinary differential equation, find Y(s). If the input to the system is doubled, i.e., the input is 2x(t) is Y(s) doubled so that its inverse Laplace transform y(t) is doubled? Is the system linear?

(c) Use MATLAB to find the solutions of the ordinary differential equation when the input is u(t) and 2u(t) with the initial conditions given above. Compare the solutions and verify your response in (b).