Verify that $y=-2 \cos (5 t)$ is a solution of $y^{\prime \prime}=-25 y$ .Suppose that y describes the motion of a mass on a spring. How does the motion start when t = 0?
A)lowest point
B)in the middle
C)highest point

Question

Quizplus · Accepted Answer

The given equation is a second-order differential equation, and the given function is a solution if it satisfies the equation. To verify, we can compute the first and second derivatives of the function and substitute them into the equation.$$y=-2 \cos (5 t)$$$$y' = 10 \sin (5 t)$$$$y'' = 50 \cos (5 t)$$Substituting these into the equation, we get:$$y''=-25 y$$$$50 \cos (5 t) = -25 (-2 \cos (5 t))$$$$50 \cos (5 t) = 50 \cos (5 t)$$The equation holds, so the given function is a solution.To determine how the motion starts when t = 0, we can evaluate the function and its first derivative at t = 0:$$y(0) = -2 \cos (5 \cdot 0) = -2$$$$y'(0) = 10 \sin (5 \cdot 0) = 0$$The function is at its lowest point when t = 0, and the mass is not moving.

Verify That y=−2cos⁡(5t)y=-2 \cos (5 t)y=−2cos(5t) Is a Solution Of y′′=−25yy^{\prime \prime}=-25 yy′′=−25y

Verify That $y=-2 \cos (5 t)$ Is a Solution Of $y^{\prime \prime}=-25 y$