Question

The principle of superposition is a useful technique in developing solutions to differential equations after an initial solution is derived. In developing such solutions it is often convenient to consider separately the response of a system to initial conditions with all forcing function inputs equal to zero, and, secondly, the response to the forcing function with all initial conditions equal to zero. Consider the following differential equation:

dy/dt + ay= bus(t)

with y(c)=0

a. Determine the solution y_i(t) to the equation for nonzero, finite values of a and c, and for b = 0.
b. Determine the solution y_u(t) of the equation for nonzero, finite values of a and b, and for c = 0.

Answer 1

A)
`y_(i)(t) = C.e^(-at)`

B)
`y_(u)(t) = - (b)/(a) e^(-at) + (b)/(a)` t > s

Explanation:

Attached below is a detailed solution of the given problem

Given equation:

dy/dt + ay =
`bu_(s)(t)` with Y(c) = 0

A) Determine the solution
`Y_(i)(t)`

`y_(i)(t) = C.e^(-at)`

B) Determine the solution
`Y_(u) (t)`

`y_(u)(t) = - (b)/(a) e^(-at) + (b)/(a)`