Q13P - Mathematical Methods in Physical Sciences

Question

Verify that ${\mathit{e}}^{\mathbf{i}\mathbf{\omega }\mathbf{t}}\mathbf{,}{\mathit{e}}^{\mathbf{-}\mathbf{i}\mathbf{\omega }\mathbf{t}}\mathbf{,}\mathit{c}\mathit{o}\mathit{s}\mathit{\omega }\mathit{t}\mathbf{,}\mathbf{ }\mathit{a}\mathit{n}\mathit{d}\mathbf{ }\mathbf{sin}\mathit{\omega }\mathit{t}$ satisfy equation (16.21).

Step-by-Step Solution

Answer

It has been verified.

Step 1: Given Information.

The given expression is $e^{\mathrm{i\omega t}},e^{-\mathrm{i\omega t}},cos\omega t, and \sin \omega t$ .

Step 2: Definition of complex series.

The numbers that are presented in the form of a + ib where, a, b are real numbers and ' i ' is an imaginary number called complex numbers.

Example: 3 + 2i .

Step 3: Use Hook’s and Newton’s Second law for testing first function..

Test the function $e^{\left(i\omega t\right)}$

$$\begin{gathered} my\text{'}=m\frac{d{y}_o{e}^{\left(i\omega t\right)}}{dt} \\ my\text{'}=im\omega y_o{e}^{\left(i\omega t\right)} \\ my\text{'}=-m{\omega }^2{y}_o{e}^{\left(i\omega t\right)} \\ my\text{'}=im\omega y_o\frac{d{e}^{\left(i\omega t\right)}}{dt} \\ my\text{'}=-m{\omega }^{2}y \end{gathered}$$

Use the Hook’s and Newton’s Second law.

$$\begin{gathered} m\frac{d^{2}y}{d{t}^2}=-ky \\ -ky=-m{\omega }^{2}y \\ {\omega }^2=\frac{k}{m} \end{gathered}$$

Step 4: Use Hook’s and Newton’s Second law for testing second function.

Test the function $e^{\left(-i\omega t\right)}$

$$\begin{gathered} my\text{'}=m\frac{d{y}_o{e}^{\left(-i\omega t\right)}}{dt} \\ my\text{'}=-im\omega y_o{e}^{\left(-i\omega t\right)} \\ my\text{'}\text{'}=-im\omega y_o\frac{d{e}^{\left(-i\omega t\right)}}{dt} \\ my\text{'}\text{'}=(-i)(-i)m{\omega }^2{y}_n{e}^{\left(i\omega t\right)} \\ my\text{'}\text{'}=-m{\omega }^{2}y \end{gathered}$$

 

Use the Hook’s and Newton’s Second law.

$$\begin{gathered} m\frac{d^{2}y}{d{t}^2}=-ky \\ -ky=-m{\omega }^{2}y \\ {\omega }^2=\frac{k}{m} \end{gathered}$$

Step 5: Use Hook’s and Newton’s Second law for testing third function.

Test the function

$$\begin{gathered} my\text{'}=m\frac{d \cos \left(\omega t\right)}{dt} \\ my\text{'}=-m{y}_e\omega \sin \left(\omega t\right) \\ my\text{'}\text{'}=-m{y}_e\omega \frac{d\sin \left(\omega t\right)}{dt} \\ my\text{'}\text{'}=-m{\omega }^2{y}_e \cos \left(\omega t\right) \\ my\text{'}\text{'}=-m{\omega }^{2}y \end{gathered}$$

 Use the Hook’s and Newton’s Second law.

$$\begin{gathered} m\frac{d^{2}y}{d{t}^2}=-ky \\ -ky=-m{\omega }^{2}y \\ {\omega }^2=\frac{k}{m} \end{gathered}$$

Step 6: Use Hook’s and Newton’s Second law for testing fourth function.

Test the function $\sin \left(\omega t\right)$

$$\begin{gathered} my\text{'}=m\frac{d\sin \left(\omega t\right)}{dt} \\ my\text{'}=m{y}_o\omega \cos \left(\omega t\right) \\ my\text{'}\text{'}=m{y}_a\omega \frac{d \cos \left(\omega t\right)}{dt} \\ my\text{'}\text{'}=-m{\omega }^2{y}_o\sin \left(\omega t\right) \\ my\text{'}\text{'}=-m{\omega }^{2}y \end{gathered}$$

 Use the Hook’s and Newton’s Second law.

 

Hence it has been shown that all the four function satisfies the equation.

$$\begin{gathered} m\frac{d^{2}y}{d{t}^2}=-ky \\ -ky=-m{\omega }^{2}y \\ {\omega }^2=\frac{k}{m} \end{gathered}$$