91Ó°ÊÓ

Determine whether the given function is homogeneous of degree zero. Rewrite those that are as functions of the single variable \(V=y / x\). $$f(x, y)=\frac{x+7}{2 y}$$

A tank initially contains \(10 \mathrm{L}\) of a salt solution. Water flows into the tank at a rate of \(3 \mathrm{L} / \mathrm{min},\) and the well-stirred mixture flows out at a rate of 2 L/min. After 5 minutes, the concentration of salt in the tank is \(0.2 \mathrm{g} / \mathrm{L} .\) Find: (a) The amount of salt in the tank initially. (b) The volume of solution in the tank when the concentration of salt is \(0.1 \mathrm{g} / \mathrm{L}\).

Verify that the given function is a solution to the given differential equation \(\left(c_{1} \text { and } c_{2}\right.\) are arbitrary constants), and state the maximum interval over which the solution is valid. $$y(x)=c_{1} e^{-5 x}+c_{2} e^{5 x}, \quad y^{\prime \prime}-25 y=0$$.

Consider the family of curves $$ x^{2}+3 y^{2}=2 c y $$ (a) Show that the differential equation of this family is \(\frac{d y}{d x}=\frac{2 x y}{x^{2}-3 y^{2}}\) (b) Determine the orthogonal trajectories to the family (1.12.5).

A glass of water whose temperature is \(50^{\circ} \mathrm{F}\) is taken outside at noon on a day whose temperature is constant at \(70^{\circ} \mathrm{F}\). If the water's temperature is \(55^{\circ} \mathrm{F}\) at \(2 \mathrm{p} . \mathrm{m} .,\) do you expect the water's temperature to reach \(60^{\circ} \mathrm{F}\) before \(4 \mathrm{p} . \mathrm{m}\). or after \(4 \mathrm{p} . \mathrm{m} . ?\) Use Newton's law of cooling to explain your answer.

Verify that the given function (or relation) defines a solution to the given differential equation and sketch some of the solution curves. If an initial condition is given, label the solution curve corresponding to the resulting unique solution. (In these problems, \(c\) denotes an arbitrary constant.) $$x^{2}+y^{2}=c, \quad y^{\prime}=-x / y$$

Consider the population model $$\frac{d P}{d t}=r(P-T) P, \quad P(0)=P_{0}$$ where \(r, T,\) and \(P_{0}\) are positive constants. (a) Perform a qualitative analysis of the differential equation in the initial-value problem (1.5.7) following the steps used in the text for the logistic equation. Identify the equilibrium solutions, the isoclines, and the behavior of the slope and concavity of the solution curves. (b) Using the information obtained in (a), sketch the slope field for the differential equation and include representative solution curves. (c) What predictions can you make regarding the behavior of the population? Consider the cases \(P_{0}T .\) The constant \(T\) is called the threshold level. Based on your predictions, why is this an appropriate term to use for \(T ?\)

Consider the RC circuit which has \(R=5 \Omega, C=\frac{1}{50}\) \(\mathrm{F},\) and \(E(t)=100 \mathrm{V} .\) If the capacitor is uncharged initially, determine the current in the circuit for \(t \geq 0\).

An RL circuit has EMF \(E(t)=10 \sin 4 t\) V. If \(R=\) \(2 \Omega, L=\frac{2}{3} \mathrm{H},\) and there is no current flowing initially, determine the current for \(t \geq 0\).

Prove that the initial-value problem $$ y^{\prime}=x \sin (x+y), \quad y(0)=1 $$ has a unique solution.