Question

Write the given third order linear equation as an equivalent system of first order equations with initial values.

4y''' - (1 + 2t^3 + 2t^2)y' - 3t^2y = 2sin(t)
With y(3) = -3, y'(3) = 0, y''(3) = 3.
use, x₁ = y
x₂ = y'
x₃ = y''

Answer 1

To write the given third order linear equation as an equivalent system of first order equations with initial values, define three variables: x₁ = y, x₂ = y', and x₃ = y''. Then, rewrite the equation using these variables and the given initial values.

Step-by-step explanation:

To write the given third order linear equation as an equivalent system of first order equations with initial values, we can define three variables: x₁ = y, x₂ = y', and x₃ = y''. Then, we can rewrite the equation as:

x₁' = x₂

x₂' = x₃

x₃' = (1 + 2t³ + 2t²)x₂/4 - 3t²x₁/4 + sin(t)/2

With the given initial values, we have x₁(3) = -3, x₂(3) = 0, and x₃(3) = 3.