Chapter 4.2, Problem 39P

(a)

To determine

To show: That the displacements $u_1\text{ and }{u}_2$ of the masses from their equilibrium positions satisfy the equations, $u_1{}^{\prime \text{}\prime }+5u_1=2u_2,u_2{}^{\prime \text{}\prime }+2u_2=2u_1$.

(b)

To determine

The general solution of the equation, $u_1{}^{\left(4\right)}+7u_1{}^{\prime \text{}\prime }+6u_1=0$.

(c)

To determine

The values of $u_1{}^{\prime \text{}\prime }\left(0\right)\text{ and }{u}_1{}^{\prime \text{}\prime \text{}\prime }\left(0\right)$ using $u_1{}^{\prime \text{}\prime }+5u_1=2u_2,u_2{}^{\prime \text{}\prime }+2u_2=2u_1$ and the initial conditions $u_1\left(0\right)=1,u_1{}^{\prime }\left(0\right)=0,u_2\left(0\right)=2,u_2{}^{\prime }\left(0\right)=0$ and to show that the solution of $u_1{}^{\left(4\right)}+7u_1{}^{\prime \text{}\prime }+6u_1=0$ satisfies all the four initial conditions given for $u_1=\cos t$ and to show that the corresponding second solution is $u_2=2\cos t$.

(d)

To determine

The values of $u_1{}^{\prime \text{}\prime }\left(0\right)\text{ and }{u}_1{}^{\prime \text{}\prime \text{}\prime }\left(0\right)$ using $u_1{}^{\prime \text{}\prime }+5u_1=2u_2,u_2{}^{\prime \text{}\prime }+2u_2=2u_1$ and the initial conditions $u_1\left(0\right)=-2,u_1{}^{\prime }\left(0\right)=0,u_2\left(0\right)=1,u_2{}^{\prime }\left(0\right)=0$ and to show that the solution of $u_1{}^{\left(4\right)}+7u_1{}^{\prime \text{}\prime }+6u_1=0$ satisfies all the four initial conditions given for $u_1=-2\cos \sqrt{6}t$ and to show that the corresponding second solution is $u_2=\cos \sqrt{6}t$.

(e)

To determine

To describe: The two distinct nodes of vibration from the sub part (c) and (d).