Question

It can be shown that {eᵗ, teᵗ} is a fundamental set of solutions of y'' – 2y' +y= 0. Determine which of the following is also a fundamental set.

A. {teᵗ, t²eᵗ}
B. {eᵗ + teᵗ, eᵗ}
C. {eᵗ – teᵗ, -eᵗ+ teᵗ}
D. {5eᵗ, 2teᵗ}
E. {eᵗ – teᵗ, eᵗ + teᵗ}
F. {-teᵗ, 5teᵗ}

Answer 1

To determine the fundamental set, we need to check if the given functions satisfy the differential equation. Option E is a fundamental set.

Step-by-step explanation:

To determine which of the sets is also a fundamental set, we need to check if the given functions satisfy the differential equation y'' - 2y' + y = 0.

Let's start with option A: {te^t, t^2e^t}. Taking the second derivative of t^2e^t, we get (2t + 2)e^t, which doesn't match with the -2y' term in the differential equation. Therefore, option A is not a fundamental set.

By using the same approach, we can also find that options B, C, D, and F are not fundamental sets.

Finally, let's check option E: {e^t - te^t, e^t + te^t}. Taking the derivatives of e^t - te^t and e^t + te^t, we find that they satisfy the differential equation. Therefore, option E is a fundamental set.