We have a second order differential equation and we have been given the general solution. Our job is to show that the solution is correct.

We do this by substituting the answer into the original 2nd order differential equation.

We need to find the second derivative of y:

y = c1 sin 2x + 3 cos 2x

First derivative:

`(dy)/(dx)=2c_1\ cos\ 2x-6\ sin\ 2x`

Second derivative:

`(d^2y)/(dx^2)=-4c_1\ sin\ 2x-12\ cos\ 2x`

Now for the check step:

`"LHS"=(d^2y)/(dx^2)+4y`

`=[-4c_1sin\ 2x-12\ cos\ 2x]+` `4(c_1sin\ 2x+3\ cos\ 2x)`

`=0`

`="RHS"`

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