91Ó°ÊÓ

Determine the quadrants in which the solution of the differential equation is an increasing function. Explain. (Do not solve the differential equation.) $$ \frac{d y}{d x}=\frac{1}{2} x^{2} y $$

Verify that the general solution satisfies the differential equation. Then find the particular solution that satisfies the initial condition. $$ \begin{aligned} &3 x^{2}+2 y^{2}=C \\ &3 x+2 y y^{\prime}=0 \\ &y=3 \text { when } x=1 \end{aligned} $$

When predicting population growth, demographers must consider birth and death rates as well as the net change caused by the difference between the rates of immigration and emigration. Let \(P\) be the population at time \(t\) and let \(N\) be the net increase per unit time resulting from the difference between immigration and emigration. So, the rate of growth of the population is given by $$ \frac{d P}{d t}=k P+N, \quad N \text { is constant. } $$ Solve this differential equation to find \(P\) as a function of time if at time \(t=0\) the size of the population is \(P_{0}\).

The management at a certain factory has found that the maximum number of units a worker can produce in a day is \(40 .\) The rate of increase in the number of units \(N\) produced with respect to time \(t\) in days by a new employee is proportional to \(40-N\) (a) Determine the differential equation describing the rate of change of performance with respect to time. (b) Solve the differential equation from part (a). (c) Find the particular solution for a new employee who produced 10 units on the first day at the factory and 19 units on the twentieth day.

Consider a tank that at time \(t=0\) contains \(v_{0}\) gallons of a solution of which, by weight, \(q_{0}\) pounds is soluble concentrate. Another solution containing \(q_{1}\) pounds of the concentrate per gallon is running into the tank at the rate of \(r_{1}\) gallons per minute. The solution in the tank is kept well stirred and is withdrawn at the rate of \(r_{2}\) gallons per minute. A 200 -gallon tank is full of a solution containing 25 pounds of concentrate. Starting at time \(t=0\), distilled water is admitted to the tank at a rate of 10 gallons per minute, and the well-stirred solution is withdrawn at the same rate. (a) Find the amount of concentrate \(Q\) in the solution as a function of \(t\). (b) Find the time at which the amount of concentrate in the tank reaches 15 pounds. (c) Find the quantity of the concentrate in the solution as \(t \rightarrow \infty\).

(a) use Euler's Method with a step size of \(h=0.1\) to approximate the particular solution of the initial value problem at the given \(x\) -value, (b) find the exact solution of the differential equation analytically, and (c) compare the solutions at the given \(x\) -value. $$ \begin{array}{lll} \text { Differential Equation } & \text { Initial Condition } & x \text { -value } \\ \frac{d y}{d x}=\frac{2 x+12}{3 y^{2}-4} & (1,2) & x=2 \end{array} $$

The rate of decomposition of radioactive radium is proportional to the amount present at any time. The half-life of radioactive radium is 1599 years. What percent of a present amount will remain after 25 years?

A calf that weighs 60 pounds at birth gains weight at the rate \(\frac{d w}{d t}=k(1200-w)\) where \(w\) is weight in pounds and \(t\) is time in years. Solve the differential equation. (a) Use a computer algebra system to solve the differential equation for \(k=0.8,0.9\), and 1 . Graph the three solutions. (b) If the animal is sold when its weight reaches 800 pounds, find the time of sale for each of the models in part (a). (c) What is the maximum weight of the animal for each of the models?

The number of bacteria in a culture is increasing according to the law of exponental growth. There are 125 bacteria in the culture after 2 hours and 350 bacteria after 4 hours. (a) Find the initial population. (b) Write an exponential growth model for the bacteria population. Let \(t\) represent time in hours. (c) Use the model to determine the number of bacteria after 8 hours. (d) After how many hours will the bacteria count be \(25,000 ?\)

The management at a certain factory has found that a worker can produce at most 30 units in a day. The learning curve for the number of units \(N\) produced per day after a new employee has worked \(t\) days is \(N=30\left(1-e^{k t}\right)\). After 20 days on the job, a particular worker produces 19 units. (a) Find the learning curve for this worker. (b) How many days should pass before this worker is producing 25 units per day?