Problem 8
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. $$ (t-1) y^{\prime \prime}-3 t y^{\prime}+4 y=\sin t, \quad y(-2)=2, \quad y^{\prime}(-2)=1 $$
Problem 8
In each of Problems 1 through 8 determine whether the given pair of functions is linearly independent or linearly dependent. \(f(x)=x^{3}, \quad g(x)=|x|^{3}\)
Problem 8
Find the general solution of the given differential equation. $$ y^{\prime \prime}-2 y^{\prime}-2 y=0 $$
Problem 8
A series circuit has a capacitor of \(0.25 \times 10^{-6}\) farad and an inductor of 1 henry. If the initial charge on the capacitor is \(10^{-6}\) coulomb and there is no initial current, find the charge \(Q\) on the capacitor at any time \(t\)
Problem 9
A mass of \(20 \mathrm{g}\) stretches a spring \(5 \mathrm{cm}\). Suppose that the mass is also attached to a viscous damper with a damping constant of \(400 \mathrm{dyne}\) -sec/cm. If the mass is pulled down an additional \(2 \mathrm{cm}\) and then released, find its position \(u\) at any time \(t .\) Plot \(u\) versus \(t .\) Determine the quasi frequency and the quasi period. Determine the ratio of the quasi period to the period of the corresponding undamped motion. Also find the time \(\tau\) such that \(|u(t)|<0.05\) \(\mathrm{cm}\) for all \(t>\tau\)
Problem 9
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. $$ t(t-4) y^{\prime \prime}+3 t y^{\prime}+4 y=2, \quad y(3)=0, \quad y^{\prime}(3)=-1 $$
Problem 9
In each of Problems 1 through 10 find the general solution of the given differential equation. \(25 y^{\prime \prime}-20 y^{\prime}+4 y=0\)
Problem 9
Find the general solution of the given differential equation. $$ u^{n}+\omega_{0}^{2} u=\cos \omega t, \quad \omega^{2} \neq \omega_{0}^{2} $$
Problem 9
The Wronskian of two functions is \(W(t)=t \sin ^{2} t .\) Are the functions linearly independent or linearly dependent? Why?
Problem 9
In each of Problems 9 through 16 find the solution of the given initial value problem. Sketch the graph of the solution and describe its behavior as \(t\) increases. $$ y^{\prime \prime}+y^{\prime}-2 y=0, \quad y(0)=1, \quad y^{\prime}(0)=1 $$
