Which of the following is a solution for the given different

Which of the following is a solution for the given differential equation? Y\" + 16y=0 y(x) = cos(8 x) -sin(8 x) y(x) = in(4 x) y(x) = sinh(2 x) y(x) = 2 sin (2 x) y(x) = cos(4 x) + 4 sin (4 x) None of the above.

Solution

Given that

y\'\' + 16y = 0

d²y/dx² + 16y = 0

D - 0perator form is,

( D² + 16)y = 0

Auxialary equation is ,

r² + 16 = 0

r² = -16

r = (-16)

r = i16 [ since , i² = -1 , i² = i ]

r = ±i4

r = 0 ± i4 [ Since , it is ± i form ]

   = 0 , = 4

If the roots are imaginary then the solution is ,

y(x) = e^x( c₁cos(x) + c₂sin(x) )

y(x) = e^0.x ( c₁cos(4x) + c₂sin(4x) )

y(x) = e⁰ ( c₁cos(4x) + c₂sin(4x) )

y(x) = c₁cos(4x) + c₂sin(4x) [ since , e⁰ = 1 ]

Therefore ,

The solution is ,

   y(x) = c₁cos(4x) + c₂sin(4x)

Option \' f \' is correct