Chapter 7

Use Euler's method with \(h=0.1\) and \(h=0.05\) to approximate \(y(1)\) and \(y(2) .\) Show the first two steps by hand. $$y^{\prime}=\sqrt{x+y}, y(0)=1$$

Find and interpret all equilibrium points for the competing species model. (Hint: There are four equilibrium points in exercise 17. $$\left\\{\begin{array}{l} x^{\prime}=0.3 x-0.2 x^{2}-0.2 x y \\ y^{\prime}=0.2 y-0.1 y^{2}-0.2 x y \end{array}\right.$$

The differential equation is separable. Find the general solution, in an explicit form if possible. Sketch several members of the family of solutions. $$y^{\prime}=\frac{x y}{1+x^{2}}$$

Involve exponential decay. The radioactive element iodine- 131 has a decay constant of -1.3863 day \(^{-1} .\) Find its half-life.

Use Euler's method with \(h=0.1\) and \(h=0.05\) to approximate \(y(1)\) and \(y(2) .\) Show the first two steps by hand. $$y^{\prime}=\sqrt{x^{2}+y^{2}}, y(0)=4$$

Find and interpret all equilibrium points for the competing species model. (Hint: There are four equilibrium points in exercise 17. $$\left\\{\begin{array}{l} x^{\prime}=0.4 x-0.3 x^{2}-0.1 x y \\ y^{\prime}=0.3 y-0.2 y^{2}-0.1 x y \end{array}\right.$$

The differential equation is separable. Find the general solution, in an explicit form if possible. Sketch several members of the family of solutions. $$y^{\prime}=\frac{2}{x y+y}$$

Involve exponential decay. The radioactive element cesium- 137 has a decay constant of -0.023 year \(^{-1} .\) Find its half-life.

Find and interpret all equilibrium points for the competing species model. (Hint: There are four equilibrium points in exercise 17. $$\left\\{\begin{array}{l} x^{\prime}=0.2 x-0.2 x^{2}-0.1 x y \\ y^{\prime}=0.1 y-0.1 y^{2}-0.2 x y \end{array}\right.$$

The differential equation is separable. Find the general solution, in an explicit form if possible. Sketch several members of the family of solutions. $$y^{\prime}=\frac{\cos ^{2} y}{4 x-3}$$

Involve exponential decay. The half-life of morphine in the human bloodstream is 3 hours. If initially there is 0.4 mg of morphine in the bloodstream, find an equation for the amount in the bloodstream at any time. When does the amount drop below \(0.01 \mathrm{mg} ?\)

Find and interpret all equilibrium points for the competing species model. (Hint: There are four equilibrium points in exercise 17. $$\left\\{\begin{array}{l} x^{\prime}=0.1 x-0.2 x^{2}-0.1 x y \\ y^{\prime}=0.3 y-0.2 y^{2}-0.1 x y \end{array}\right.$$

The differential equation is separable. Find the general solution, in an explicit form if possible. Sketch several members of the family of solutions. $$y^{\prime}=\frac{\left(y^{2}+1\right) \ln x}{4 y}$$

The population models in exercises 17-22 are competing species models. Suppose that \(x(t)\) and \(y(t)\) are the populations of two species of animals that compete for the same plant food. Explain why the interaction terms for both species are negative.

Solve the IVP, explicitly if possible. $$y^{\prime}=3(x+1)^{2} y, y(0)=1$$

Involve exponential decay. Strontium-90 is a dangerous radioactive isotope. Because of its similarity to calcium, it is easily absorbed into human bones. The half-life of strontium- 90 is 28 years. If a certain amount is absorbed into the bones due to exposure to a nuclear explosion, what percentage will remain after 50 years?

Solve the IVP, explicitly if possible. $$y^{\prime}=\frac{x-1}{y^{2}}, y(0)=2$$

Involve exponential decay.The half-life of uranium \(^{235} \mathrm{U}\) is approximately \(0.7 \times 10^{9}\) years. If 50 grams are buried at a nuclear waste site, how much will remain after 100 years?

Find the equilibrium solutions and determine which are stable and which are unstable. $$y^{\prime}=2 y-y^{2}$$

Solve the IVP, explicitly if possible. $$y^{\prime}=\frac{4 x^{2}}{y}, y(0)=2$$

Involve exponential decay. Scientists dating a fossil estimate that \(20 \%\) of the original amount of carbon-14 is present. Recalling that the half-life is 5730 years, approximately how old is the fossil?

Find the equilibrium solutions and determine which are stable and which are unstable. $$y^{\prime}=y^{3}-1$$

Solve the IVP, explicitly if possible. $$y^{\prime}=\frac{x-1}{y}, y(0)=-2$$

Find the equilibrium solutions and determine which are stable and which are unstable. $$y^{\prime}=y^{2}-y^{4}$$

Solve the IVP, explicitly if possible. $$y^{\prime}=\frac{4 y}{x+3}, y(-2)=1$$

Find the equilibrium solutions and determine which are stable and which are unstable. $$y^{\prime}=e^{-y}-1$$

Solve the IVP, explicitly if possible. $$y^{\prime}=\frac{3 x}{4 y+1}, y(1)=4$$

Find the equilibrium solutions and determine which are stable and which are unstable. $$y^{\prime}=(1-y) \sqrt{1+y^{2}}$$

Solve the IVP, explicitly if possible. $$y^{\prime}=\frac{4 x}{\cos y}, y(0)=0$$

Find the equilibrium solutions and determine which are stable and which are unstable. $$y^{\prime}=\sqrt{1-y^{2}}$$

Solve the IVP, explicitly if possible. $$y^{\prime}=\frac{\tan y}{x}, y(1)=\frac{\pi}{2}$$

Write the second-order equation as a system of first-order equations. $$y^{\prime \prime}-(\cos x) y^{\prime}+x y^{2}=2 x$$

Use equation ( 2.6 ) to help solve the IVP. $$y^{\prime}=3 y(2-y), y(0)=1$$

Involve Newton's Law of Cooling. At 10: 07 P.M. you find a secret agent murdered. Next to him is a martini that got shaken before the secret agent could stir it. Room temperature is \(70^{\circ} \mathrm{F}\). The martini warms from \(60^{\circ} \mathrm{F}\) to \(61^{\circ} \mathrm{F}\) in the 2 minutes from 10: 07 P.M. to 10: 09 P.M. If the secret agent's martinis are always served at \(40^{\circ} \mathrm{F}\), what was the time of death?

Many species of trees are plagued by sudden infestations of worms. Let \(x(t)\) be the population of a species of worm on a particular tree. For some species, a model for population change is \(x^{\prime}=0.1 x(1-x / k)-x^{2} /\left(1+x^{2}\right)\) for some positive constant \(k .\) If \(k=10,\) show that there is only one positive equilibrium solution. If \(k=50,\) show that there are three positive equilibrium solutions. Sketch the direction field for \(k=50 .\) Explain why the middle equilibrium value is called a threshold. An outbreak of worms corresponds to crossing the threshold for a large value of \(k\) ( \(k\) is determined by the resources available to the worms).

Use equation ( 2.6 ) to help solve the IVP. $$y^{\prime}=y(3-y), y(0)=2$$

Involve Newton's Law of Cooling. Twenty minutes after being served a cup of fast-food coffee, it is still too hot to drink at \(160^{\circ} \mathrm{F}\). Two minutes later, the temperature has dropped to \(158^{\circ} \mathrm{F}\). Your friend, whose coffee is also too hot to drink, speculates that since the temperature is dropping an average of 1 degree per minute, it was served at \(180^{\circ} \mathrm{F} .\) Explain what is wrong with this logic. Was the actual serving temperature hotter or cooler than \(180^{\circ} \mathrm{F} ?\)

Apply Euler's method with \(h=0.1\) to the initial value problem \(y^{\prime}=y^{2}-1, y(0)=3\) and estimate \(y(0.5) .\) Repeat with \(h=0.05\) and \(h=0.01 .\) In general, Euler's method is more accurate with smaller \(h\) -values. Conjecture how the exact solution behaves in this example. (This is explored further in exercises \(34-36 .)\)

Use equation ( 2.6 ) to help solve the IVP. $$y^{\prime}=2 y(5-y), y(0)=4$$

Use equation ( 2.6 ) to help solve the IVP. $$y^{\prime}=y(2-y), y(0)=1$$

Graph the solution of \(y^{\prime}=y^{2}-1, y(0)=3,\) given in exercise \(34 .\) Find an equation of the vertical asymptote. Explain why Euler's method would be "unaware" of the existence of this asymptote and would therefore provide very unreliable approximations.

Use equation ( 2.6 ) to help solve the IVP. $$y^{\prime}=y(1-y), y(0)=\frac{3}{4}$$

Involve compound interest. If you invest \(\$ 1000\) at an annual interest rate of \(8 \%,\) compare the value of the investment after 1 year under the following forms of compounding: annual, monthly, daily, continuous.

Use equation ( 2.6 ) to help solve the IVP. $$y^{\prime}=y(3-y), y(0)=0$$

Involve compound interest. Person A invests \(\$ 10,000\) in 1990 and person \(\mathrm{B}\) invests \(\$ 20,000\) in \(2000 .\) If both receive \(12 \%\) interest (compounded continuously), what are the values of the investments in \(2010 ?\)

One of the authors bought a set of basketball trading cards in 1985 for \(\$ 34 .\) In \(1995,\) the "book price" for this set was \(\$ 9800 .\) Assuming a constant percentage return on this investment, find an equation for the worth of the set at time \(t\) years (where \(t=0\) corresponds to 1985 ). At this rate of return, what would the set have been worth in \(2005 ?\)

Find all equilibrium points. $$\left\\{\begin{array}{l} x^{\prime}=(x-y)(1-x-y) \\ y^{\prime}=2 x-x y \end{array}\right.$$

Find all equilibrium points. $$\left\\{\begin{array}{l}x^{\prime}=(2+x)(y-x) \\\ y^{\prime}=(4-x)(x+y)\end{array}\right.$$

In \(1975,\) income between \(\$ 16,000\) and \(\$ 20,000\) was taxed at \(28 \% .\) In \(1988,\) income between \(\$ 16,000\) and \(\$ 20,000\) was taxed at \(15 \% .\) This makes it seem as if taxes went down considerably between 1975 and \(1988 .\) Taking inflation into account, briefly explain why this is not a valid comparison.

Find all equilibrium points. $$\left\\{\begin{array}{l}x^{\prime}=-x+y \\\ y^{\prime}=y+x^{2}\end{array}\right.$$