Question

Approximate the following second-order differential equation with a difference equation.

d²y(t)/ dt²+a1dy(t)//dt+a0y(t)=x(t)

Answer 1

To approximate the given second-order differential equation with a difference equation, we can use the Central Difference Approximation formula. By applying this formula and rearranging terms, we can obtain the difference equation.

Step-by-step explanation:

To approximate the given second-order differential equation with a difference equation, we can use the Central Difference Approximation formula. First, let's denote y(t), dy(t)/dt, and d²y(t)/dt² as y, y1, and y2, respectively. Then, the equation becomes:

y2 + a1y1 + a0y = x

Now, we can approximate the derivatives using the Central Difference Approximation: y' = (y(t+h) - y(t-h))/(2h) and y'' = (y(t+h) - 2y(t) + y(t-h))/h².

Substituting these approximations into the original equation, we get:

(y(t+h) - 2y(t) + y(t-h))/h² + a1(y(t+h) - y(t-h))/(2h) + a0y(t) = x(t)

By rearranging terms and solving for y(t+h), we obtain the difference equation:

y(t+h) = (2h²x(t) - 2h²a0y(t) + 2h²a1y(t-h) + h²y(t-h) + h²y(t+h))/(h² + 2ha1)