Chapter 3

In each of Problems 1 through 8 determine whether the given pair of functions is linearly independent or linearly dependent. $f(t)=t,g(t)=t^{-1}$

In each of Problems 1 through 10 find the general solution of the given differential equation. $y^{\prime \prime }-6y^{\prime }+9y=0$

A mass of $100\mathrm{g}$ stretches a spring $5\mathrm{cm}$. If the mass is set in motion from its equilibrium position with a downward velocity of $10\mathrm{cm}/\sec ,$ and if there is no damping, determine the position $u$ of the mass at any time $t.$ When does the mass first return to its equilibrium position?

use Euler’s formula to write the given expression in the form a + ib. $${\pi }^{-1+2i}$$

Find the general solution of the given differential equation. In Problems 11 and $12\mathrm{g}$ is an arbitrary continuous function. $$y^{\prime \prime }+4y^{\prime }+4y=t^{-2}{e}^{-2t},t>0$$

determine the longest interval in which the given initial value problem is certain to have a unique twice differentiable solution. Do not attempt to find the solution. $$t{y}^{\prime \prime }+3y=t,y(1)=1,y^{\prime }(1)=2$$

A mass weighing 3 Ib stretches a spring 3 in. If the mass is pushed upward, contracting the spring a distance of 1 in, and then set in motion with a downward velocity of $2\mathrm{ft}$ sec, and if there is no damping, find the position $u$ of the mass at any time $t.$ Determine the frequency, period, amplitude, and phase of the motion.

In each of Problems 1 through 10 find the general solution of the given differential equation. $4y^{\prime \prime }+17y^{\prime }+4y=0$

In each of Problems 1 through 8 determine whether the given pair of functions is linearly independent or linearly dependent. $f(t)=3t,g(t)=|t|$

In each of Problems 1 through 8 determine whether the given pair of functions is linearly independent or linearly dependent. $f(x)=x^3,g(x)=|x{|}^3$