91Ó°ÊÓ

Find general solutions of the differential equations. Primes denote derivatives with respect to \(x\) throughout. $$ (x+y) y^{\prime}=1 $$

In Problems 17 through 26, first verify that \(y(x)\) satisfies the given differential equation. Then determine a value of the constant \(C\) so that \(y(x)\) satisfies the given initial condition. Use a computer or graphing calculator (if desired) to sketch several typical solutions of the given differential equation, and highlight the one that satisfies the given initial condition. \(y^{\prime}+3 x^{2} y=0 ; y(x)=C e^{-x^{3}}, y(0)=7\)

Find explicit particular solutions of the initial value problems $$ (\tan x) \frac{d y}{d x}=y, \quad y\left(\frac{1}{2} \pi\right)=\frac{1}{2} \pi $$

Solve the differential equations in Problem by regarding \(y\) as the independent variable rather than \(x\). $$ \left(1-4 x y^{2}\right) \frac{d y}{d x}=y^{3} $$

Find a general solution and any singular solutions of the differential equation dymyslashd \(x=y \sqrt{y^{2}-1}\). Determine the points \((a, b)\) in the plane for which the initial value problem \(y^{\prime}=y \sqrt{y^{2}-1}, y(a)=b\) has (a) no solution, (b) a unique solution, (c) infinitely many solutions.

(Population growth) A certain city had a population of 25000 in 1960 and a population of 30000 in 1970 . Assume that its population will continue to grow exponentially at a constant rate. What population can its city planners expect in the year \(2000 ?\)

(Population growth) In a certain culture of bacteria, the number of bacteria increased sixfold in \(10 \mathrm{~h}\). How long did it take for the population to double?

The half-life of radioactive cobalt is \(5.27\) years. Suppose that a nuclear accident has left the level of cobalt radiation in a certain region at 100 times the level acceptable for human habitation. How long will it be until the region is again habitable? (Ignore the probable presence of other radioactive isotopes.)

A certain moon rock was found to contain equal numbers of potassium and argon atoms. Assume that all the argon is the result of radioactive decay of potassium (its half-life is about \(1.28 \times 10^{9}\) years) and that one of every nine potassium atom disintegrations yields an argon atom. What is the age of the rock, measured from the time it contained only potassium?

Early one morning it began to snow at a constant rate. At 7 A.M. a snowplow set off to clear a road. By 8 A.M. it had traveled 2 miles, but it took two more hours (until \(10 \mathrm{~A}, \mathrm{M}\).) for the snowplow to go an additional 2 miles. (a) Let \(t=0\) when it began to snow and let \(x\) denote the distance traveled by the snowplow at time \(t\). Assuming that the snowplow clears snow from the road at a constant rate (in cubic feet per hour, say), show that $$ k \frac{d x}{d t}=\frac{1}{t} $$