Question

Find the solution to the following second-order-system. Provide the solution in the time domain y(t). Use the Laplace tables uploaded on Canvas to help you with the inverse transform back into the time domain.

y¨​+3y˙​+y=1
y(0)=0
y˙​(0)​=1​

Answer 1

To find the solution to the given second-order system, first find the characteristic equation, then find the roots of the equation, and finally use the roots to write the solution in the time domain using the initial conditions.

Step-by-step explanation:

To find the solution to the given second-order system and provide it in the time domain, we can start by finding the characteristic equation:

Characteristic equation: s^2 + 3s + 1 = 0

Next, we can find the roots of the characteristic equation using the quadratic formula:

Roots: s = (-3 ± sqrt(5))/2

Now, we can use the roots to write the solution in the time domain:

y(t) = C1 * e^((-3 + sqrt(5))/2 * t) + C2 * e^((-3 - sqrt(5))/2 * t) + 1

Finally, we can use the initial conditions y(0) = 0 and y'(0) = 1 to solve for the constants C1 and C2.