Problem 84
Use Euler's Method with \(n=5\) steps over the interval \(t=[0,1] .\) Then solve the initial-value problem exactly. How close is your Euler's Method estimate? $$ y^{\prime}=3^{x}-2 y, y(0)=0 $$
Problem 84
Substitute \(y=a e^{t} \cos t+b e^{t} \sin t\) into \(y^{\prime}=2 e^{t} \cos t\) to find a particular solution.
Problem 85
Solve \(y^{\prime}=e^{k t}\) with the initial condition \(y(0)=0\) and solve \(y^{\prime}=1\) with the same initial condition. As \(k\) approaches 0, what do you notice?
Problem 85
A car drives along a freeway, accelerating according to \(a=5 \sin (\pi t)\), where \(t\) represents time in minutes. Find the velocity at any time \(t\), assuming the car starts with an initial speed of \(60 \mathrm{mph}\).
Problem 86
You throw a ball of mass 2 kilograms into the air with an upward velocity of \(8 \mathrm{~m} / \mathrm{s}\). Find exactly the time the ball will remain in the air, assuming that gravity is given by \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\).
Problem 88
You drop the same ball of mass 5 kilograms out of the same airplane window at the same height, except this time you assume a drag force proportional to the ball's velocity, using a proportionality constant of 3 and the ball reaches terminal velocity. Solve for the distance fallen as a function of time. How long does it take the ball to reach the ground?
Problem 92
The human population (in thousands) of Nevada in 1950 was roughly 160 . If the carrying capacity is estimated at 10 million individuals, and assuming a growth rate of \(2 \%\) per year, develop a logistic growth model and solve for the population in Nevada at any time (use 1950 as time \(=0\) ). What population does your model predict for 2000 ? How close is your prediction to the true value of \(1,998,257\) ?
