Question

Verify by substitution that the given functions are solutions of the given differential equation. Note that any primes denote derivatives with respect to x.

y"+16y = 0, y₁ = cos 4x, y₂ = sin 4x

Answer 1

To verify that y₁ = cos 4x and y₂ = sin 4x are solutions of the differential equation y"+16y = 0, we need to substitute them into the equation and verify if the equation holds true.

Step-by-step explanation:

To verify that y₁ = cos 4x and y₂ = sin 4x are solutions of the differential equation y"+16y = 0, we need to substitute them into the equation and check if it holds true.

For y₁ = cos 4x, we have y"₁ = -16cos 4x. Substitute y₁ and y"₁ into the differential equation:

-16cos 4x + 16cos 4x = 0. Therefore, y₁ = cos 4x is a solution.

For y₂ = sin 4x, we have y"₂ = -16sin 4x. Substitute y₂ and y"₂ into the differential equation:

-16sin 4x + 16sin 4x = 0. Therefore, y₂ = sin 4x is also a solution.