Question

y = c1 cos(3x) + c2 sin(3x) is a two-parameter family of solutions of the second-order DE y'' + 9y = 0. If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions. (If not possible, enter IMPOSSIBLE.)

Answer 1

Complete Question

The complete question is shown on the first uploaded image

Answer:

The value is
`y  =  5sin3x`

Explanation:

From the question we are told that

`y =  c_1\ cos3x + c_2 \  sin 3x`

given that y(0) = 0

We have that

`c_1\ cos3(0) + c_2 \  sin 3(0)  =  0`

=>
`c_1 =  0`

This means that

`y  =  c_2 \  sin 3x`

We also given that

`y[(\pi)/(6) ] =  5`

So

`c_2  sin (3 *  (\pi)/(6) )  =  5`

=>
`c_2  =  5`

Hence

`y  =  5sin3x`