91Ó°ÊÓ

Use the existence and uniqueness theorem to prove that \(y(x)=3\) is the only solution to the initial-value problem $$ y^{\prime}=\frac{x}{x^{2}+1}\left(y^{2}-9\right), \quad y(0)=3 $$

Determine the current flowing in an RL circuit if the applied EMF is \(E(t)=E_{0} \sin \omega t,\) where \(E_{0}\) and \(\omega\) are constants. Identify the transient part of the solution and the steady-state solution.

According to data from the U.S. Bureau of the Census, the population (measured in millions of people)of the U.S. in 1950, 1960, and 1970 was, respectively, 151.3, 179.4, and 203.3. (a) Using the 1950 and 1960 population figures, solve the corresponding Malthusian population model. (b) Determine the logistic model corresponding to the given data. (c) On the same set of axes, plot the solution curves obtained in (a) and (b). From your plots, determine the values the different models would have predicted for the population in 1980 and 1990 , and compare these predictions to the actual values of 226.54 and 248.71 , respectively.

For Problems (a) Determine all equilibrium solutions. (b) Determine the regions in the \(x y\) -plane where the solutions are increasing, and determine those regions where they are decreasing. (c) Determine the regions in the \(x y\) -plane where the solution curves are concave up, and determine those regions where they are concave down. (d) Sketch representative solution curves in each region of the \(x y\) -plane identified in (c). $$y^{\prime}=(y+2)(y-1)$$

Sketch the slope field and some representative solution curves for the given differential equation. $$y^{\prime}=1 / x$$

The pressure \(p,\) and density, \(\rho,\) of the atmosphere at a height \(y\) above the earth's surface are related by $$ d p=-g \rho d y $$ Assuming that \(p\) and \(\rho\) satisfy the adiabatic equation of state \(p=p_{0}\left(\frac{\rho}{\rho_{0}}\right)^{\gamma},\) where \(\gamma \neq 1\) is a constant and \(p_{0}\) and \(\rho_{0}\) denote the pressure and density at the earth's surface, respectively, show that $$ p=p_{0}\left[1-\frac{(\gamma-1)}{\gamma} \cdot \frac{\rho_{0} g y}{p_{0}}\right]^{\gamma /(\gamma-1)} $$.

Solve the differential equation for Newton's law of cooling by viewing it as a first-order linear differential equation.

An object is released from rest at a height of 100 meters above the ground. Neglecting frictional forces, the subsequent motion is governed by the initial-value problem $$ \frac{d^{2} y}{d t^{2}}=g, \quad y(0)=0, \quad \frac{d y}{d t}(0)=0 $$where \(y(t)\) denotes the displacement of the object from its initial position at time \(t\). Solve this initial-value problem and use your solution to determine the time when the object hits the ground.

A five-foot-tall boy tosses a tennis ball straight up from the level of the top of his head. Neglecting frictional forces, the subsequent motion is governed by the differential equation $$ \frac{d^{2} y}{d t^{2}}=g $$ If the object hits the ground 8 seconds after the boy releases it, find (a) the time when the tennis ball reaches its maximum height. (b) the maximum height of the tennis ball.

A pyrotechnic rocket is to be launched vertically upwards from the ground. For optimal viewing, the rocket should reach a maximum height of 90 meters above the ground. Ignore frictional forces. (a) How fast must the rocket be launched in order to achieve optimal viewing? (b) Assuming the rocket is launched with the speed determined in part (a), how long after the rocket is launched will it reach its maximum height?