Question

Compute the solution to x +2x +2x =0. For x0 = 0 mm, v0 = 1 mm/s and write down the closed-form expression for the response.

Answer 1

The answer is "
`\bold{e^(-t) \sin t}`"

Step-by-step explanation:

Given equation:

`\to x+2x+2x=0.............(a)`

Let x= e^{mt} be a solution of the equation will be:

`\to m^2+2m+2=0`

compare the value with the
`am^2+bm+c=0`

`\to a= 1\\\to b= -2\\\to c= 2\\`

Formula:

`\bold{=(-b \pm √(b^2-4ac))/(2a)}\\\\=(-2 \pm √((-2)^2-4* 1 * 2))/(2 * 1)\\\\=(-2 \pm √(4-8))/(2)\\\\=(-2 \pm √(-4))/(2)\\\\=(-2 \pm 2i)/(2)\\\\`

Calculate the g.s at equation (a):

`\to x = e^(-t) (c_1 \cos t+ c_2 \sin t)............(b)\\\\\to x = e^(-t) (c_2 \cos t -c_1 \sin t) - e^(-t) (c_1 \cos t + c_2\sin t) ...............(c)\\\\`

`_(when) \\\\\to t= 0\\\\\to x=x_0\\\\\to x= v_0 =1 \\\\ \ then \ form \ equation \ (b) \ and \ equation \ (c)`

`\to a= 0 \\ \to c_2 -c_1 =1 \\ \to c_2=1`

The value of
`x = e^(-t) ( 0 * \cos t + 1 * \sin t)`

`\boxed{\bold{x = e^(-t) (\sin t)}}`