Question

Verify by substitution whetherthe given functions are solutions of the given DE. Primes denote derivatives with respect to x.y!! +y′= cos 2x;y"= cosx+sinx,y # = cos 2x,y$ =sin 2x

Answer 1

Complete Question

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Answer:

`y_1` is not a solution of the differential equation

`y_2` is not a solution of the differential equation

`y_3` is not a solution of the differential equation

Explanation:

The differential equation given is
`y'' + y' = cos2x`

Let consider the first equation to substitute

`y_1 = cosx +sinx`

`y_1' = -sinx +cosx`

`y_1'' = -cosx -sinx`

So

`y_1'' - y_1' = -cosx -sinx -sinx +cosx`

`y_1'' + y_1' = -2sinx`

So

`-2sinx \\e cos2x`

This means that
`y_1` is not a solution of the differential equation

Let consider the second equation to substitute

`y_2 = cos2x`

`y_2' = -2sin2x`

`y_2'' = -4cos2x`

So

`y_2'' + y_2' = -4cos2x-2sin2x`

So

`-4cos2x-2sin2x \\e cos2x`

This means that
`y_2` is not a solution of the differential equation

Let consider the third equation to substitute

`y_3 = sin 2x`

`y_3' =  2cos 2x`

`y_3'' =  -4sin2x`

So

`y_3'' + y_3' = -4sin2x - 2cos2x`

So

`-4sin2x - 2cos2x \\e cos2x`

This means that
`y_3` is not a solution of the differential equation