A $500$-gal aquarium is cleansed by the recirculating filter system schematically. Water containing impurities is pumped out at a rate of $15 \mathrm{gal} / \mathrm{min}$, filtered, and returned to the aquarium at the same rate. Assume that passing through the filter reduces the concentration of impurities by a fractional amount $\alpha$, as shown in the figure. In other words, if the impurity concentration upon entering the filter is $c(t)$, the exit concentration is $\alpha c(t)$, where $0<\alpha<1$. (a) Apply the basic conservation principle (rate of change = rate in - rate out) to obtain a differential equation for the amount of impurities present in the aquarium at time t. Assume that filtering occurs instantaneously. If the outflow concentration at any time is $c(t)$, assume that the inflow concentration at that same instant is $\alphac(t)$. (b) What value of filtering constant a will reduce impurity levels to $1%$ of their original values in a period of $3$ hr?

A certain drug is being administered intravenously to a hospital patient. Fluid containing 5mg/cm^3 of the drug enters the patient's bloodstream at a rate of 100 cm^3/h. The drug is absorbed by body tissues or otherwise leaves the bloodstream at a rate proportional to the amount present, with a rate constant of 0.4 (h)^(-1). (a) Assuming that the drug is always uniformly distributed throughout the bloodstream, write a differential equation for the amount of the drug that is present in the bloodstream at any time. (b) How much of the drug is present in the bloodstream after a long time?

Collar $A$ is connected as shown to a 50 -lb load and can slide on a frictionless horizontal rod. Determine the distance $x$ for which the collar is in equilibrium when $P=48 \mathrm{lb}$.

Collar $A$ is connected as shown to a 50-lb load and can slide on a frictionless horizontal rod. Determine the distance $x$ for which the collar is in equilibrium when $P=48\ \mathrm{lb}$.

Collar $A$ is connected as shown to a $50-\mathrm{lb}$ load and can slide on a frictionless horizontal rod. Determine the magnitude of the force $\mathbf{P}$ required to maintain the equilibrium of the collar when $(a) x=4.5 \ \text{in.}$, (b) $x=15 \mathrm{in}$

Two cables are tied together at $C$ as shown. Find the value of a for which the tension is as small as possible (a) in cable $BC$, (b) in both cables simultaneously. In each case determine the tension in both cables.

If the differential equation is linear, determine the general solution. If the differential equation is nonlinear, obtain an implicit solution (and an explicit solution if possible). If an initial condition is given, solve the initial value problem.

$$\sqrt{t} y^{\prime}-\sqrt{y} t=0$$

(a) Obtain an implicit solution and, if possible, an explicit solution of the initial value problem. (b) If you can find an explicit solution of the problem, determine the $t$-interval of existence.

$$\frac{d y}{d t}=e^{t-y}, y(0)=1$$

Find the general solution.

$$t y^{\prime}+4 y=0$$

Find the general solution of the differential equation specified.

$$\frac{dy}{dt} = (ty)^{2}$$

Solve by factoring. $t^{2}+3 t+2=0$

Use the Runge-Kutta method to approximate x(0.2) and y(0.2). Compare the results for h=0.2 and h=0.1. $x^{\prime}=-y+t \\ y^{\prime}=x-t \\ x(0)=-3, y(0)=5$

Use the Laplace transform to solve the initial value problem.

$$y^{\prime}+2 y=26 \sin 3 t, y(0)=3$$

solve the initial value problem y'+2y=g(t),y(0)=0, where g(t)=1, 0≤t≤1,0,t>1