A system described by the following differential equation   + 3  +2𝑦 = x(𝑡) is initially at rest. For input x(𝑡) = 2𝑢(𝑡), the output y(t) isSelect one:a. (1 − 2𝑒 −𝑡 + 𝑒 −2 ) 𝑢(𝑡)  b. (0.5 + 𝑒 −𝑡 + 1.5𝑒 −2 ) 𝑢(𝑡)c. (1 + 2𝑒 −𝑡 − 2𝑒 −2𝑡 ) 𝑢(𝑡)d. (0.5 + 2𝑒 −𝑡 + 2𝑒 −2𝑡 ) 𝑢(𝑡)

To solve the given differential equation, we can use the Laplace transform. The Laplace transform of the differential equation is:

sY(s) + 3sY(s) + 2Y(s) = X(s)

where Y(s) and X(s) are the Laplace transforms of y(t) and x(t) respectively.

Since the system is initially at rest, we can assume y(0) = 0. Taking the Laplace transform of the input x(t) = 2u(t), where u(t) is the unit step function, we get:

X(s) = 2/s

Substituting the Laplace transforms into the differential equation, we have:

sY(s) + 3sY(s) + 2Y(s) = 2/s

Combining like terms, we get:

Y(s)(s + 3s + 2) = 2/s

Simplifying further, we have:

Y(s)(s^2 + 3s + 2) = 2/s

Factoring the denominator, we get:

Y(s)(s + 1)(s + 2) = 2/s

Now, we can solve for Y(s) by isolating it on one side of the equation:

Y(s) = 2/(s(s + 1)(s + 2))

Using partial fraction decomposition, we can express Y(s) as:

Y(s) = A/s + B/(s + 1) + C/(s + 2)

To find the values of A, B, and C, we can multiply both sides of the equation by the denominator and equate the coefficients of the corresponding powers of s. After solving for A, B, and C, we get:

A = 1 B = -2 C = 1

Substituting these values back into the equation for Y(s), we have:

Y(s) = 1/s - 2/(s + 1) + 1/(s + 2)

Taking the inverse Laplace transform of Y(s), we can find the output y(t):

y(t) = 1 - 2e^(-t) + e^(-2t)

Therefore, the correct answer is option c. (1 + 2e^(-t) - 2e^(-2t))u(t).