Question

Find the frequency ω for which the particular solution to the differential equation 3(d²y​/dt²)+dy​/dt+2y=eᶦωᵗ has the largest amplitude. You can assume a positive frequency ω>0. Probably the easiest way to do this is to find the particular solution in the form Aᵉᶦωᵗ and then minimize the modulus of the denominator of A over all frequencies ω.

Answer 1

The frequency ω for the particular solution to the given differential equation to have the largest amplitude is ω = √(2/3).

Step-by-step explanation

To find the frequency ω that maximizes the amplitude of the particular solution, we'll start by assuming the particular solution takes the form A * e^(iωt). Substituting this into the given differential equation, we get:

`\[3 (d²y)/(dt²) + (dy)/(dt) + 2y = e^(iωt)\]`

Replace
`\(y = A e^(iωt)\)`into the equation:

`\[3iω(3Aiωe^(iωt)) + iω(Ae^(iωt)) + 2Ae^(iωt) = e^(iωt)\]`

Simplify by dividing through by
`\(e^(iωt)\):`

`\[3iω(3Aiω) + iω(A) + 2A = 1\]`

Solving for A:

`\[9ω²Ai + ωAi + 2A = 1\]`

`\[A(9ω²i + ωi + 2) = 1\]`

`\[A = (1)/(9ω²i + ωi + 2)\]`

We want to maximize the amplitude, which is the modulus of A. The amplitude is maximized when the denominator
`\(9ω²i + ωi + 2\)` is minimized. To minimize the modulus of the denominator, consider the denominator's square:

`\[(9ω²i + ωi + 2)(9ω²i + ωi + 2)^* = |9ω²i + ωi + 2|^2\]`

Where
`\(^*\)`denotes the complex conjugate. The denominator's square becomes:

`\[81ω^4 + 18ω² + 4\]`

To minimize this expression, take its derivative with respect to ω and set it to zero:

`\[(d)/(dω)(81ω^4 + 18ω² + 4) = 0\]`

`\[324ω^3 + 36ω = 0\]`

`\[ω(9ω² + 1) = 0\]`

Solving for ω, we get
`\(ω = \sqrt{(2)/(3)}\),`which is the frequency that maximizes the amplitude of the particular solution.