1.5: Problems

1. Find all of the solutions of the first order differential equations. When an initial condition is given, find the particular solution satisfying that condition.
   1. \(\dfrac{d y}{d x}=\dfrac{e^{x}}{2 y}\)
   2. \(\dfrac{d y}{d t}=y^{2}\left(1+t^{2}\right), y(0)=1\).
   3. \(\dfrac{d y}{d x}=\dfrac{\sqrt{1-y^{2}}}{x}\).
   4. \(x y^{\prime}=y(1-2 y), \quad y(1)=2\).
   5. \(y^{\prime}-(\sin x) y=\sin x\).
   6. \(x y^{\prime}-2 y=x^{2}, y(1)=1\).
   7. \(\dfrac{d s}{d t}+2 s=s t^{2}, \quad, s(0)=1\).
   8. \(x^{\prime}-2 x=t e^{2 t}\).
   9. \(\dfrac{d y}{d x}+y=\sin x, y(0)=0\).
   10. \(\dfrac{d y}{d x}-\dfrac{3}{x} y=x^{3}, y(1)=4\).
2. For the following determine if the differential equation is exact. If it is not exact, find the integrating factor. Integrate the equations to obtain solutions.
   1. \(\left(3 x^{2}+6 x y^{2}\right) d x+\left(6 x^{2} y+4 y^{3}\right) d y=0\)
   2. \(\left(x+y^{2}\right) d x-2 x y d y=0\).
   3. \((\sin x y+x y \cos x y) d x+x^{2} \cos x y d y=0\).
   4. \(\left(x^{2}+y\right) d x-x d y=0\).
   5. \(\left(2 x y^{2}-3 y^{3}\right) d x+\left(7-3 x y^{2}\right) d y=0\).
3. Consider the differential equation

\(\dfrac{d y}{d x}=\dfrac{x}{y}-\dfrac{x}{1+y}\)

5. Find the I-parameter family of solutions (general solution) of this equation.
6. Find the solution of this equation satisfying the initial condition \(y(0)=1\). Is this a member of the i-parameter family?

4. A ball is thrown upward with an initial velocity of \(49 \mathrm{~m} / \mathrm{s}\) from \(539 \mathrm{~m}\) high. How high does the ball get and how long does in take before it hits the ground? [Use results from the simple free fall problem, \(\left.y^{\prime \prime}=-g .\right]\)
5. Consider the case of free fall with a damping force proportional to the velocity, \(f_{D}=\pm k v\) with \(k=0.1 \mathrm{~kg} / \mathrm{s}\).
   1. Using the correct sign, consider a \(50 \mathrm{~kg}\) mass falling from rest at a height of room. Find the velocity as a function of time. Does the mass reach terminal velocity?
   2. Let the mass be thrown upward from the ground with an initial speed of \(50 \mathrm{~m} / \mathrm{s}\). Find the velocity as a function of time as it travels upward and then falls to the ground. How high does the mass get? What is its speed when it returns to the ground?
6. An piece of a satellite falls to the ground from a height of \(10,000 \mathrm{~m}\). Ignoring air resistance, find the height as a function of time. [Hint: For free fall from large distances.

\(\ddot{h}=-\dfrac{G M}{(R+h)^{2}}\)

Multiplying both sides by \(\dot{h}\), show that

\(\dfrac{d}{d t}\left(\dfrac{1}{2} \dot{h}^{2}\right)=\dfrac{d}{d t}\left(\dfrac{G M}{R+h}\right)\)

Integrate and solve for \(\dot{h}\). Further integrating gives \(h(t).]\)

7. The problem of growth and decay is stated as follows: The rate of change of a quantity is proportional to the quantity. The differential equation for such a problem is

\(\dfrac{d y}{d t}=\pm k y\)

The solution of this growth and decay problem is \(y(t)=y_{0} e^{\pm k t}\). Use this solution to answer the following questions if forty percent of a radioactive substance disappears in 100 years.

1. What is the half-life of the substance?
2. After how many years will \(90 \%\) be gone?

8. Uranium 237 has a half-life of \(6.78\) days. If there are \(10.0\) grams of U-237 now, then how much will be left after two weeks?
9. The cells of a particular bacteria culture divide every three and a half hours. If there are initially 250 cells, how many will there be after ten hours?
10. The population of a city has doubled in 25 years. How many years will it take for the population to triple?
11. Identify the type of differential equation. Find the general solution and plot several particular solutions. Also, find the singular solution if one exists.
    1. \(y=x y^{\prime}+\dfrac{1}{y^{\prime}}\).
    2. \(y=2 x y^{\prime}+\ln y^{\prime}\).
    3. \(y^{\prime}+2 x y=2 x y^{2}\).
    4. \(y^{\prime}+2 x y=y^{2} e^{x^{2}}\).
12. Find the general solution of the Riccati equation given the particular solution(A function \(F(x, y)\) is said to be homogeneous of degree \(k\) if \(F(t x, t y)=t^{k} F(x, y)\).)
    1. \(x y^{\prime}-y^{2}+(2 x+1) y=x^{2}+2 x, y_{1}(x)=x\).
    2. \(y^{\prime} e^{-x}+y^{2}-2 y e^{x}=1-e^{2 x}, y_{1}(x)=e^{x}\).
13. The initial value problem

\(\dfrac{d y}{d x}=\dfrac{y^{2}+x y}{x^{2}}, \quad y(1)=1\)

does not fall into the class of problems considered in this chapter. The function on the right-hand side is a homogeneous function of degree zero. However, if one substitutes \(y(x)=x z(x)\) into the differential equation, one obtains an equation for \(z(x)\) which can be solved. Use this substitution to solve the initial value problem for \(y(x)\).

14. If \(M(x, y)\) and \(N(x, y)\) are homogeneous functions of the same degree, then \(M / N\) can be written as a function of \(y / x\). This suggests that a substitution of \(y(x)=x z(x)\) into \(M(x, y) d x+N(x, y) d y\) might simplify the equation. For the following problems use this method to find the family of solutions.
    1. \(\left(x^{2}-x y+y^{2}\right) d x-x y d y=0\)
    2. \(x y d x-\left(x^{2}+y^{2}\right) d y=0\).
    3. \(\left(x^{2}+2 x y-4 y^{2}\right) d x-\left(x^{2}-8 x y-4 y^{2}\right) d y=0\).
15. Find the family of orthogonal curves to the given family of curves.
    1. \(y=a x\)
    2. \(y=a x^{2}\).
    3. \(x^{2}+y^{2}=2 a x\).
16. The temperature inside your house is \(70^{\circ} \mathrm{F}\) and it is \(30^{\circ} \mathrm{F}\) outside. At \(1:\) oo A.M. the furnace breaks down. At 3:00 A.M. the temperature in the house has dropped to \(50^{\circ} \mathrm{F}\). Assuming the outside temperature is constant and that Newton’s Law of Cooling applies, determine when the temperature inside your house reaches \(40^{\circ} \mathrm{F}\).
17. A body is discovered during a murder investigation at 8:oo P.M. and the temperature of the body is \(70^{\circ} \mathrm{F}\). Two hours later the body temperature has dropped to \(60^{\circ} \mathrm{F}\) in a room that is at \(50^{\circ} \mathrm{F}\). Assuming that Newton’s Law of Cooling applies and the body temperature of the person was \(98.6^{\circ} \mathrm{F}\) at the time of death, determine when the murder occurred.
18. Newton’s Law of Cooling states that the rate of heat loss of an object is proportional to the temperature gradient, or

\(\dfrac{d Q}{d t}=h A \Delta T\)

where \(Q\) is the thermal energy, \(h\) is the heat transfer coefficient, \(A\) is the surface area of the body, and \(\Delta T=T-T_{a}\). If \(Q=C T\), where \(C\) is the heat capacity, then we recover Equation \(1.3.7\) with \(k=h A / C\).

However, there are modifications which include convection or radiation. Solve the following models and compare the solution behaviors.

1. Newton \(T^{\prime}=-k\left(T-T_{a}\right)\)
2. Dulong-Petit \(T^{\prime}=-k\left(T-T_{a}\right)^{5 / 4}\)
3. Newton-Stefan \(T^{\prime}=-k\left(T-T_{a}\right)-\epsilon \sigma\left(T^{4}-T_{a}^{4}\right) \approx-k\left(T-T_{a}\right)-\) \(b\left(T-T_{a}\right)^{2}\).

19. Initially a 200 gallon tank is filled with pure water. At time \(t=0\) a salt concentration with 3 pounds of salt per gallon is added to the container at the rate of 4 gallons per minute, and the well-stirred mixture is drained from the container at the same rate.
    1. Find the number of pounds of salt in the container as a function of time.
    2. How many minutes does it take for the concentration to reach 2 pounds per gallon?
    3. What does the concentration in the container approach for large values of time? Does this agree with your intuition?
    4. Assuming that the tank holds much more than 200 gallons, and everything is the same except that the mixture is drained at 3 gallons per minute, what would the answers to parts a and b become?
20. You make two gallons of chili for a party. The recipe calls for two teaspoons of hot sauce per gallon, but you had accidentally put in two tablespoons per gallon. You decide to feed your guests the chili anyway. Assume that the guests take i cup/min of chili and you replace what was taken with beans and tomatoes without any hot sauce. [ 1 gal \(=16\) cups and \(1 \mathrm{~Tb}=3\) tsp.]
    1. Write down the differential equation and initial condition for the amount of hot sauce as a function of time in this mixture-type problem.
    2. Solve this initial value problem.
    3. How long will it take to get the chili back to the recipe’s suggested concentration?