Questions: Find the value(s) of ω for which y=cos ωt satisfies d²y/dt²+9y=0 Enter your answers in increasing order. ω=i ω=1

Transcript text: Find the value(s) of $\omega$ for which $y=\cos \omega t$ satisfies \[ \frac{d^{2} y}{d t^{2}}+9 y=0 \] Enter your answers in increasing order. \[ \begin{array}{l} \omega=\mathbf{i} \\ \omega=1 \end{array} \]

Solution

Solution Steps

To find the value(s) of $\omega$ for which $y = \cos(\omega t)$ satisfies the differential equation $\frac{d^2 y}{d t^2} + 9y = 0$, we need to:

Compute the second derivative of $y$ with respect to $t$.
Substitute $y$ and its second derivative into the differential equation.
Solve the resulting equation for $\omega$.

Step 1: Define the Function and Compute the Second Derivative

Given the function $ y = \cos(\omega t) $, we compute the second derivative with respect to $ t $: \[ \frac{d^2 y}{d t^2} = -\omega^2 \cos(\omega t) \]

Step 2: Substitute into the Differential Equation

Substitute $ y $ and $ \frac{d^2 y}{d t^2} $ into the differential equation: \[ \frac{d^2 y}{d t^2} + 9y = 0 \] \[ -\omega^2 \cos(\omega t) + 9 \cos(\omega t) = 0 \]

Step 3: Simplify the Equation

Factor out $\cos(\omega t)$ from the equation: \[ \cos(\omega t) (-\omega^2 + 9) = 0 \] Since $\cos(\omega t) \neq 0$, we have: \[ -\omega^2 + 9 = 0 \]

Step 4: Solve for $\omega$

Solve the equation for $\omega$: \[ \omega^2 = 9 \] \[ \omega = \pm 3 \]