To solve for the equation of motion for the given second-order linear differential equation with damping:

$y''(t) + 6y'(t) + ky(t) = 0$ with initial conditions: $y(0) = 1, \quad y'(0) = 0$

Step 1: Find the characteristic equation The characteristic equation for this differential equation is obtained by assuming solutions of the form $y(t) = e^{rt}$. Substituting this into the differential equation yields:

$r^2 + 6r + k = 0$

For $k = 7$, the characteristic equation becomes:

$r^2 + 6r + 7 = 0$

Step 2: Solve the characteristic equation We can solve the characteristic equation using the quadratic formula:

$r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For our equation $r^2 + 6r + 7 = 0$, $a = 1$, $b = 6$, and $c = 7$. Substituting these values into the quadratic formula:

$r = \frac{-6 \pm \sqrt{6^2 - 4(1)(7)}}{2(1)} = \frac{-6 \pm \sqrt{36 - 28}}{2} = \frac{-6 \pm \sqrt{8}}{2} = \frac{-6 \pm 2\sqrt{2}}{2}$

This simplifies to:

$r = -3 \pm \sqrt{2}$

Thus, the roots are complex and given by:

$r_1 = -3 + \sqrt{2}, \quad r_2 = -3 - \sqrt{2}$

Step 3: Write the general solution Since the roots are complex, the general solution to the differential equation is of the form:

$y(t) = e^{-3t} \left( C_1 \cos(\sqrt{2}t) + C_2 \sin(\sqrt{2}t) \right)$

Step 4: Apply initial conditions

We are given $y(0) = 1$ and $y'(0) = 0$.

At $t = 0$:

$y(0) = e^{-3(0)} \left( C_1 \cos(0) + C_2 \sin(0) \right) = 1 \Rightarrow C_1 = 1$

Now, differentiate the solution to find $y'(t)$:

$y'(t) = e^{-3t} \left( -3C_1 \cos(\sqrt{2}t) - \sqrt{2}C_1 \sin(\sqrt{2}t) - 3C_2 \sin(\sqrt{2}t) + \sqrt{2}C_2 \cos(\sqrt{2}t) \right)$

At $t = 0$, applying the initial condition $y'(0) = 0$:

$y'(0) = e^{-3(0)} \left( -3C_1 \cos(0) + \sqrt{2}C_2 \cos(0) \right) = 0 \Rightarrow -3C_1 + \sqrt{2}C_2 = 0$

Substitute $C_1 = 1$:

$-3 + \sqrt{2}C_2 = 0 \Rightarrow C_2 = \frac{3}{\sqrt{2}}$

Step 5: Write the final solution

Thus, the equation of motion for $k = 7$ is:

$y(t) = e^{-3t} \left( \cos(\sqrt{2}t) + \frac{3\sqrt{2}}{2} \sin(\sqrt{2}t) \right)$

This is the required equation of motion for $k = 7$.

Would you like help with the equations for $k = 9$ and $k = 11$, or with any other part of the problem?