Questions: Determine whether the function (y=8 sin (2 x)) is a solution of the differential equation (y^prime prime prime-8 y=0). Yes No

Solution Steps

Let \( y = 8 \sin(2x) \).

The first derivative of \( y \) is given by: \[ y' = \frac{dy}{dx} = 16 \cos(2x) \]

The second derivative of \( y \) is: \[ y'' = \frac{d^2y}{dx^2} = -32 \sin(2x) \]

The third derivative of \( y \) is: \[ y''' = \frac{d^3y}{dx^3} = -64 \cos(2x) \]

Substituting \( y \) and \( y''' \) into the differential equation \( y''' - 8y = 0 \): \[ -64 \cos(2x) - 8(8 \sin(2x)) = -64 \cos(2x) - 64 \sin(2x) \]

The left-hand side of the equation simplifies to: \[ -64 \cos(2x) - 64 \sin(2x} \] This expression does not equal zero for all \( x \). Therefore, the function \( y = 8 \sin(2x) \) does not satisfy the differential equation \( y''' - 8y = 0 \).

Final Answer