12.5: Exercises

12.1. Water draining form a container. In Example 12.3, we verified that the function \(h(t)=\left(\sqrt{h_{0}}-k \frac{t}{2}\right)^{2}\) is a solution to the differential equation (12.2). Based on the meaning of the problem, for how long does this solution remain valid?

12.2. Verifying a solution. Verify that the function \(y(t)=1-\left(1-y_{0}\right) e^{-t}\) satisfies the initial value problem (differential equation and initial condition) (12.6).

12.3. Linear differential equation. Consider the differential equation

\[\frac{d y}{d t}=a-b y \nonumber \]

where \(a, b\) are constants.

(a) Show that the function

\[y(t)=\frac{a}{b}-C e^{-b t} \nonumber \]

satisfies the above differential equation for any constant \(C\).

(b) Show that by setting

\[C=\frac{a}{b}-y_{0} \nonumber \]

we also satisfy the initial condition

\[y(0)=y_{0} . \nonumber \]

Remark: you have shown that the function

\[y(t)=\left(y_{0}-\frac{a}{b}\right) e^{-b t}+\frac{a}{b} \nonumber \]

is a solution to the initial value problem (i.e differential equation plus initial condition)

\[\frac{d y}{d t}=a-b y, \quad y(0)=y_{0} . \nonumber \]

12.4. Complete the steps. Complete the algebraic steps to show that the solution to Equation (12.4) can be obtained by the substitution \(z(t)=\) \(a-b y(t)\).

12.5. Verifying a solution. Show that the function

\[y(t)=\frac{1}{1-t} \nonumber \]

is a solution to the differential equation and initial condition

\[\frac{d y}{d t}=y^{2}, \quad y(0)=1 . \nonumber \]

Comment on what happens to this solution as \(t\) approaches 1 .

12.6. Verifying solutions. For each of the following, show the given function \(y\) is a solution to the given differential equation.

(a) \(t \cdot \frac{d y}{d t}=3 y, y=2 t^{3}\).

(b) \(\frac{d^{2} y}{d t^{2}}+y=0, y=-2 \sin t+3 \cos t\).

(c) \(\frac{d^{2} y}{d t^{2}}-2 \frac{d y}{d t}+y=6 e^{t}, y=3 t^{2} e^{t}\).

12.7. Verifying a solution. Show the function determined by the equation \(2 x^{2}+x y-y^{2}=C\), where \(C\) is a constant and \(2 y \neq x\), is a solution to the differential equation \((x-2 y) \frac{d y}{d x}=-4 x-y\).

12.8. Determining the constant. Find the constant \(C\) that satisfies the given initial conditions.

(a) \(2 x^{2}-3 y^{2}=C,\left.y\right|_{x=0}=2\).

(b) \(y=C_{1} e^{5 t}+C_{2} t e^{5 t},\left.y\right|_{t=0}=1\) and \(\left.\frac{d y}{d t}\right|_{t=0}=0\).

(c) \(y=C_{1} \cos \left(t-C_{2}\right),\left.y\right|_{t=\frac{\pi}{2}}=0\) and \(\left.\frac{d y}{d t}\right|_{t=\frac{\pi}{2}}=1\).

12.9. Friction and terminal velocity. The velocity of a falling object changes due to the acceleration of gravity, but friction has an effect of slowing down this acceleration. The differential equation satisfied by the velocity \(v(t)\) of the falling object is

\[\frac{d v}{d t}=g-k v \nonumber \]

where \(g\) is acceleration due to gravity and \(k\) is a constant that represents the effect of friction. An object is dropped from rest from a plane.

(a) Find the function \(v(t)\) that represents its velocity over time.

(b) What happens to the velocity after the object has been falling for a long time (but before it has hit the ground)?

12.10. Alcohol level. Alcohol enters the blood stream at a constant rate \(k \mathrm{gm}\) per unit time during a drinking session. The liver gradually converts the alcohol to other, non-toxic byproducts. The rate of conversion per unit time is proportional to the current blood alcohol level, so that the differential equation satisfied by the blood alcohol level is

\[\frac{d c}{d t}=k-s c \nonumber \]

where \(k, s\) are positive constants. Suppose initially there is no alcohol in the blood.

Find the blood alcohol level \(c(t)\) as a function of time from \(t=0\), when the drinking started.

12.11. Checking a solution. Check that the differential equation (12.7) has the right sign, so that a hot object cools off in a colder environment.

12.12. Details of Newtons Law of Cooling. Fill in the missing steps in the solution to Newton’s Law of Cooling in Example 12.5.

12.13. Newton’s Law of Cooling. Newton’s Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between the temperature of the object, \(T\), and the ambient (environmental) temperature, \(E\). This leads to the differential equation

\[\frac{d T}{d t}=k(E-T) \nonumber \]

where \(k>0\) is a constant that represents the material properties and, \(E\) is the ambient temperature. (We assume that \(E\) is also constant.)

(a) Show that the function

\[T(t)=E+\left(T_{0}-E\right) e^{-k t} \nonumber \]

which represents the temperature at time \(t\) satisfies this equation.

(b) The time of death of a murder victim can be estimated from the temperature of the body if it is discovered early enough after the crime has occurred.

Suppose that in a room whose ambient temperature is \(E=20^{\circ} \mathrm{C}\), the temperature of the body upon discovery is \(T=30^{\circ} \mathrm{C}\), and that a second measurement, one hour later is \(T=25^{\circ} \mathrm{C}\).

Determine the approximate time of death.

Remark: use the fact that just prior to death, the temperature of the victim was \(37^{\circ} \mathrm{C}\).

12.14. A cup of coffee. The temperature of a cup of coffee is initially 100 degrees C. Five minutes later, \((t=5)\) it is 50 degrees \(\mathrm{C}\). If the ambient temperature is \(A=20\) degrees \(\mathrm{C}\), determine how long it takes for the temperature of the coffee to reach 30 degrees \(\mathrm{C}\).

12.15. Newton’s Law of Cooling applied to data. The data presented in Table \(12.4\) was gathered in producing Figure \(2.2\) for cooling milk during yoghurt production. According to Newton’s Law of Cooling, this data can be described by the formula

\[T=E+(T(0)-E) e^{-k t} . \nonumber \]

where \(T(t)\) is the temperature of the milk (in degrees Fahrenheit) at time \(t\) (in min), \(E\) is the ambient temperature, and \(k\) is some constant that we determine in this exercise.

Table 12.4: Cooling milk data for Exercise 12.5.15 .

time (min)	Temp
\(0.0\)	\(190.0\)
\(0.5\)	\(185.5\)
\(1.0\)	\(182.0\)
\(1.5\)	\(179.2\)
\(2.0\)	\(176.0\)
\(2.5\)	\(172.9\)
\(3.0\)	\(169.5\)
\(3.5\)	\(167.0\)
\(4.0\)	\(164.6\)
\(4.5\)	\(162.2\)
\(5.0\)	\(159.8\)

(a) Rewrite this relationship in terms of the quantity \(Y(t)=\ln (T(t)-\) \(E)\), and show that \(Y(t)\) is related linearly to the time \(t\).

(b) Explain how the constant \(k\) could be found from this converted form of the relationship.

(c) Use the data in the table and your favourite spreadsheet (or similar software) to show that the data so transformed appears to be close to linear. Assume that the ambient temperature was \(E=20^{\circ} \mathrm{F}\).

(d) Use the same software to determine the constant \(k\) by fitting a line to the transformed data.

12.16. Infant weight gain. During the first year of its life, the weight of a baby is given by

\[y(t)=\sqrt{3 t+64} \nonumber \]

where \(t\) is measured in some convenient unit.

(a) Show that \(y\) satisfies the differential equation

\[\frac{d y}{d t}=\frac{k}{y} \nonumber \]

where \(k\) is some positive constant.

(b) What is the value for \(k\) ?

(c) Suppose we adopt this differential equation as a model for human growth. State concisely (that is, in one sentence) one feature about this differential equation which makes it a reasonable model. State one feature which makes it unreasonable.

12.17. Lake Fishing. Fish Unlimited is a company that manages the fish population in a private lake. They restock the lake at constant rate (to restock means to add fish to the lake): \(N\) fishers are allowed to fish in the lake per day. The population of fish in the lake, \(F(t)\) is found to satisfy the differential equation

\[\frac{d F}{d t}=I-\alpha N F \nonumber \]

(a) At what rate are fish added per day according to Equation (12.18)? Give both value and units.

(b) What is the average number of fish caught by one fisher? Give both the value and units.

(c) What is being assumed about the fish birth and mortality rates in Equation (12.18)?

(d) If the fish input and number of fishers are constant, what is the steady state level of the fish population in the lake?

(e) At time \(t=0\) the company stops restocking the lake with fish. Give the revised form of the differential equation (12.18) that takes this into account, assuming the same level of fishing as before. How long would it take for the fish to fall to \(25 \%\) of their initial level?

(f) When the fish population drops to the level \(F_{l o w}\), fishing is stopped and the lake is restocked with fish at the same constant rate (Eqn (12.18), with \(\alpha=0\).) Write down the revised version of Equation (12.18) that takes this into account. How long would it take for the fish population to double?

12.18. Tissue culture. Cells in a tissue culture produce a cytokine (a chemical that controls the growth of other cells) at a constant rate of 10 nano-Moles per hour \((\mathrm{nM} / \mathrm{h})\). The chemical has a half-life of 20 hours.

Give a differential equation (DE) that describes this chemical production and decay. Solve this DE assuming that at \(t=0\) there is no cytokine. \(\left[1 \mathrm{nM}=10^{-9} \mathrm{M}\right]\).

12.19. Glucose solution in a tank. A tank that holds 1 liter is initially full of plain water. A concentrated solution of glucose, containing \(0.25 \mathrm{gm} / \mathrm{cm}^{3}\) is pumped into the tank continuously, at the rate \(10 \mathrm{~cm}^{3} / \mathrm{min}\) and the mixture (which is continuously stirred to keep it uniform) is pumped out at the same rate.

How much glucose is in the tank after 30 minutes? After a long time? (hint: write a differential equation for \(c\), the concentration of glucose in the tank by considering the rate at which glucose enters and the rate at which glucose leaves the tank.)

12.20. Pollutant in a lake. A lake of constant volume \(V\) gallons contains \(Q(t)\) pounds of pollutant at time \(t\) evenly distributed throughout the lake. Water containing a concentration of \(k\) pounds per gallon of pollutant enters the lake at a rate of \(r\) gallons per minute, and the well-mixed solution leaves at the same rate.

(a) Set up a differential equation that describes the way that the amount of pollutant in the lake changes.

(b) Determine what happens to the pollutant level after a long time if this process continues.

(c) If \(k=0\) find the time \(T\) for the amount of pollutant to be reduced to one half of its initial value.

12.21. A sugar solution. Sugar dissolves in water at a rate proportional to the amount of sugar not yet in solution. Let \(Q(t)\) be the amount of sugar undissolved at time \(t\). The initial amount is \(100 \mathrm{~kg}\) and after 4 hours the amount undissolved is \(70 \mathrm{~kg}\).

(a) Find a differential equation for \(Q(t)\) and solve it.

(b) How long does it take for \(50 \mathrm{~kg}\) to dissolve?

12.22. Leaking water tank. A cylindrical tank with cross-sectional area \(A\) has a small hole through which water drains. The height of the water in the \(\operatorname{tank} y(t)\) at time \(t\) is given by:

\[y(t)=\left(\sqrt{y_{0}}-\frac{k t}{2 A}\right)^{2} \nonumber \]

where \(k, y_{0}\) are constants.

(a) Show that the height of the water, \(y(t)\), satisfies the differential equation

\[\frac{d y}{d t}=-\frac{k}{A} \sqrt{y} . \nonumber \]

(b) What is the initial height of the water in the tank at time \(t=0\) ?

(c) At what time is the tank be empty?

(d) At what rate is the volume of the water in the tank changing when \(t=0\) ?

12.23. Determining constants. Find those constants \(a, b\) so that \(y=e^{x}\) and \(y=e^{-x}\) are both solutions of the differential equation

\[y^{\prime \prime}+a y^{\prime}+b y=0 . \nonumber \]

12.24. Euler’s method. Solve the decay equation in Example (12.11) analytically, that is, find the formula for the solution in terms of a decaying exponential, and then compare your values to the approximate solution values \(y_{1}\) and, \(y_{2}\) computed with Euler’s method.

12.25. Comparing approximate and true solutions:

(a) Use Euler’s method to find an approximate solution to the differential equation

\[\frac{d y}{d x}=y \nonumber \]

with \(y(0)=1\). Use a step size \(h=0.1\) and find the values of \(y\) up to \(x=0.5\). Compare the value you have calculated for \(y(0.5)\) using Euler’s method with the true solution of this differential equation. What is the error i.e. the difference between the true solution and the approximation?

(b) Now use Euler’s method on the differential equation

\[\frac{d y}{d x}=-y \nonumber \]

with \(y(0)=1\). Use a step size \(h=0.1\) again and find the values of \(y\) up to \(x=0.5\). Compare the value you have calculated for \(y(0.5)\) using Euler’s method with the true solution of this differential equation. What is the error this time?

12.26. Beginning Euler’s method. Give the first 3 steps of Euler’s method for the problem in Example 12.13.

12.27. Euler’s method and a spreadsheet. Use the spreadsheet and Euler’s method to solve the differential equation shown below:

\[d y / d t=0.5 y(2-y) \nonumber \]

Use a step size of \(h=0.1\) and show (on the same graph) solutions for the following four initial values:

\[y(0)=0.5, y(0)=1, y(0)=1.5, y(0)=2.25 \nonumber \]

For full credit, include a short explanation your process (e.g. 1-2 sentences and whatever equations you implemented on the spreadsheet.)