Chapter 3

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

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

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 \(10 \mathrm{cm} / \mathrm{sec},\) 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. $$ \pi^{-1+2 i} $$

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

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. $$ t y^{\prime \prime}+3 y=t, \quad y(1)=1, \quad y^{\prime}(1)=2 $$

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. \(4 y^{\prime \prime}+17 y^{\prime}+4 y=0\)

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

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