29.10. EXERCISES 571

11. Although the gas supply was shut off, the air in the building continued to circulate.When the gas was shut off, the temperature in the building was 70 and after fivehours, the temperature had fallen to a chilly 50 degrees. If the outside temperaturewas at 10 degrees, what is the constant in Newton’s law of cooling?

12. A radioactive substance decays according to how much is present. Thus the equationis A′ =−kA. If after 40 years, there is 5/6 of the amount initially there still present,what is the half life of this substance?

13. You have the following initial value problem y′+ y = sin t, y(0) = y0. Letting y bethe solution to this initial value problem, find a function u(t) which does not dependon y0 and limt→∞ |y(t)−u(t)|= 0.

14. A pond which holds V cubic meters is being polluted at the rate of 10+ sin(2πt) kgper year. The periodic source represents seasonal variability. The total volume of thelake is constant because it loses 1

4V cubic meters per year and gains the same. Aftera long time, what is the average amount of pollutant in this lake in a year?

Bernouli Equations

15. Solve the following initial value problems involving Bernouli equations.

(a) y′+2xy = xy3, y(1) = 2

(b) y′+ sin(t)y = sin(t)y2, y(1) = 1

(c) y′+2y = x2y3, y(1) =−1

(d) y′−2x3y = x3y−1, y(1) = 1

(e) y′+ y = x2y−2, y(1) = 1

(f) y′+ x3y = x3y−2, y(1) =−1

16. Consider y′ = py− qy2,y(0) = pmq where p,q are positive and m > 1. Solve this

Bernouli equation and also find limt→∞ y(t).

17. Consider y′ = 3y− y3, y(0) = 1. Solve this Bernouli equation and find limt→∞ y(t).

18. Find the solution to the Bernouli equation y′ = (cos t +1)y− y3, y(0) = 1. Hint:You may have to leave the solution in terms of an integral.

19. Actually the drag force of a small object moving through the air is proportional notto the speed but to the square of the speed. Thus a falling object would satisfy thefollowing equation for downward velocity. v′ = g− kv2. Here g is acceleration ofgravity in whatever units are desired. Find limt→∞ v(t) in terms of g,k. Hint: Lookat the equation.

20. A Riccati equation is like a Bernouli equation except you have an extra functionadded in. These are of the form y′ = a(t)+ b(t)y+ c(t)y2. If you have a solution,y1, show that y(t) = y1 (t)+ 1

v(t) will be another solution provided v satisfies a suit-able first order linear equation. Thus the set of all such y will involve a constantof integration and so can be regarded as a general solution to the Riccati equation.These equations result in a very natural way when you consider y′ = f (t,y) and ap-proximate f (t,y) by fixing t and approximating the resulting function of y with asecond order Taylor polynomial.

Separable Equations

21. Solve the following initial value problems involving separable equations. The or-dered pair given is to be included in the solution curve.