Problem 6
In each of Problems 1 through 8 determine whether the given pair of functions is linearly independent or linearly dependent. \(f(t)=t, \quad g(t)=t^{-1}\)
Problem 6
In each of Problems 1 through 10 find the general solution of the given differential equation. \(y^{\prime \prime}-6 y^{\prime}+9 y=0\)
Problem 6
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?
Problem 6
use Euler’s formula to write the given expression in the form a + ib. $$ \pi^{-1+2 i} $$
Problem 7
Find the general solution of the given differential equation. In Problems 11 and \(12 \mathrm{g}\) is an arbitrary continuous function. $$ y^{\prime \prime}+4 y^{\prime}+4 y=t^{-2} e^{-2 t}, \quad t>0 $$
Problem 7
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. $$ t y^{\prime \prime}+3 y=t, \quad y(1)=1, \quad y^{\prime}(1)=2 $$
Problem 7
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.
Problem 7
In each of Problems 1 through 10 find the general solution of the given differential equation. \(4 y^{\prime \prime}+17 y^{\prime}+4 y=0\)
Problem 7
In each of Problems 1 through 8 determine whether the given pair of functions is linearly independent or linearly dependent. \(f(t)=3 t, \quad g(t)=|t|\)
Problem 8
In each of Problems 1 through 8 determine whether the given pair of functions is linearly independent or linearly dependent. \(f(x)=x^{3}, \quad g(x)=|x|^{3}\)
