91Ó°ÊÓ

Show that each function \(y=f(x)\) is a solution of the accompanying differential equation. \(2 y^{\prime}+3 y=e^{-x}\) a. \(y=e^{-x}\) b. \(y=e^{-x}+e^{-(3 / 2) x}\) c. \(y=e^{-x}+C e^{-(3 / 2) x}\)

Show that each function \(y=f(x)\) is a solution of the accompanying differential equation. $$y=\frac{1}{x} \int_{1}^{x^{\prime}} \frac{e^{t}}{t} d t, \quad x^{2} y^{\prime}+x y=e^{x}$$

In some chemical reactions, the rate at which the amount of a substance changes with time is proportional to the amount present. For the change of \(\delta\) -glucono lactone into gluconic acid, for example, $$\frac{d y}{d t}=-0.6 y$$ when \(t\) is measured in hours. If there are 100 grams of \(\delta\) -glucono lactone present when \(t=0,\) how many grams will be left after the first hour?

The intensity \(L(x)\) of light \(x\) feet beneath the surface of the ocean satisfies the differential equation $$\frac{d L}{d x}=-k L$$ As a diver, you know from experience that diving to \(18 \mathrm{ft}\) in the Caribbean Sea cuts the intensity in half. You cannot work without artificial light when the intensity falls below one-tenth of the surface value. About how deep can you expect to work without artificial light?

Suppose that electricity is draining from a capacitor at a rate that is proportional to the voltage \(V\) across its terminals and that, if \(t\) is measured in seconds, $$\frac{d V}{d t}=-\frac{1}{40} V$$ Solve this equation for \(V\), using \(V_{0}\) to denote the value of \(V\) when \(t=0 .\) How long will it take the voltage to drop to \(10 \%\) of its original value?

Suppose that the bacteria in a colony can grow unchecked, by the law of exponential change. The colony starts with 1 bacterium and doubles every half- hour. How many bacteria will the colony contain at the end of 24 hours? (Under favorable laboratory conditions, the number of cholera bacteria can double every 30 min. In an infected person, many bacteria are destroyed, but this example helps explain why a person who feels well in the morning may be dangerously ill by evening.)

A colony of bacteria is grown under ideal conditions in a laboratory so that the population increases exponentially with time. At the end of 3 hours there are 10,000 bacteria. At the end of 5 hours there are \(40,000 .\) How many bacteria were present initially?

Suppose the amount of oil pumped from one of the canyon wells in Whittier, California, decreases at the continuous rate of \(10 \%\) per year. When will the well's output fall to one-fifth of its present value?

The half-life of the plutonium isotope is 24,360 years. If \(10 \mathrm{g}\) of plutonium is released into the atmosphere by a nuclear accident, how many years will it take for \(80 \%\) of the isotope to decay?

Suppose that a cup of soup cooled from \(90^{\circ} \mathrm{C}\) to \(60^{\circ} \mathrm{C}\) after 10 min in a room whose temperature was \(20^{\circ} \mathrm{C} .\) Use Newton's Law of Cooling to answer the following questions. a. How much longer would it take the soup to cool to \(35^{\circ} \mathrm{C} ?\) b. Instead of being left to stand in the room, the cup of \(90^{\circ} \mathrm{C}\) soup is put in a freezer whose temperature is \(-15^{\circ} \mathrm{C}\). How long will it take the soup to cool from \(90^{\circ} \mathrm{C}\) to \(35^{\circ} \mathrm{C} ?\)