Question

Rewrite the following differential equation as an equivalent system of first-order differential equations. Use the variables

x₁ = .x,, x₂ = x', x₃ = x" etc.
Equation to rewrite: x⁽⁴⁾ + 17x" - 17' + 17x = -9 cos(2t)

Answer 1

The given equation can be rewritten as a system of first-order differential equations with x₁ = x, x₂ = x', x₃ = x''.

Explanation:

The given equation is a fourth-order differential equation, which can be rewritten as a system of first-order differential equations. To do this, we will need to introduce new variables. We will use x₁ = x, x₂ = x', x₃ = x'' as our variables, where x' is the first derivative of x with respect to time, and x'' is the second derivative of x with respect to time.

We can now rewrite the given equation as a system of equations:

x₁' = x₂

x₂' = x₃

x₃' = -9cos(2t) - 17x₃ + 17x₂ - 17x₁

This system of equations is an equivalent representation of the given fourth-order differential equation. It allows us to solve for the derivatives of x by breaking up the problem into smaller parts. This is a common technique used to solve higher-order differential equations.