Chapter 1

A boy 2 meters tall shoots a toy rocket straight up from head level at 10 meters per second. Assume the acceleration of gravity is 9.8 meters/sec \(^{2}\). (a) What is the highest point above the ground reached by the rocket? (b) When does the rocket hit the ground?

Solve the given differential equation. $$y^{\prime \prime}-2 y^{\prime}=6 e^{3 x}$$

Use Euler's method with the specified step size to determine the solution to the given initial-value problem at the specified point. $$y^{\prime}=4 y-1, \quad y(0)=1, \quad h=0.05, \quad y(0.5)$$.

Determine whether the given differential equation is exact. $$y e^{x y} d x+\left(2 y-x e^{x y}\right) d y=0$$

Solve the given differential equation. $$\frac{d y}{d x}+y=4 e^{x}$$

A tank initially contains 600 L of solution in which there is dissolved 1500 grams of chemical. A solution containing \(5 \mathrm{g} / \mathrm{L}\) of the chemical flows into the tank at a rate of \(6 \mathrm{L} / \mathrm{min},\) and the well-stirred mixture flows out at a rate of 3 L/min. Determine the concentration of chemical in the tank after one hour.

Determine whether the given function is homogeneous of degree zero. Rewrite those that are as functions of the single variable \(V=y / x\). $$f(x, y)=\frac{5 x+2 y}{9 x-4 y}$$

The number of bacteria in a culture grows at a rate that is proportional to the number present. Initially there were 10 bacteria in the culture. If the doubling time of the culture is 3 hours, find the number of bacteria that were present after 24 hours.

Solve the given differential equation. $$\frac{d y}{d x}=2 x y$$

Determine whether the differential equation is linear or nonlinear. $$\frac{d^{2} y}{d x^{2}}+e^{x} \frac{d y}{d x}=x^{2}$$.

Determine the differential equation giving the slope of the tangent line at the point \((x, y)\) for the given family of curves. $$y=c e^{2 x}$$

Determine the order of the differential equation. $$\frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+y=e^{x}$$

A racquetball player standing at the back wall of the court hits the ball from a height of 2 feet horizontally toward the front wall at 80 miles per hour. The length of a regulation racquetball court is 40 feet. Does the ball reach the front wall before hitting the ground? Neglect air resistance, and assume the acceleration of gravity is 32 feet/sec \(^{2}\).

Solve the given differential equation. $$y^{\prime \prime}=2 x^{-1} y^{\prime}+4 x^{2}$$

Use Euler's method with the specified step size to determine the solution to the given initial-value problem at the specified point. $$y^{\prime}=-\frac{2 x y}{1+x^{2}}, \quad y(0)=1, \quad h=0.1, \quad y(1)$$.

Determine whether the given differential equation is exact. $$[\cos (x y)-x y \sin (x y)] d x-x^{2} \sin (x y) d y=0$$

Solve the given differential equation. $$\frac{d y}{d x}=\frac{y^{2}}{x^{2}+1}$$

A container initially contains \(10 \mathrm{L}\) of water in which there is \(20 \mathrm{g}\) of salt dissolved. A solution containing \(4 \mathrm{g} / \mathrm{L}\) of salt is pumped into the container at a rate of \(2 \mathrm{L} / \mathrm{min},\) and the well-stirred mixture runs out at a rate of 1 L/min. How much salt is in the tank after \(40 \min ?\)

The number of bacteria in a culture grows at a rate that is proportional to the number present. After 10 hours, there were 5000 bacteria present, and after 12 hours, there were 6000 bacteria present. Determine the initial size of the culture and the doubling time of the population.

Determine the differential equation giving the slope of the tangent line at the point \((x, y)\) for the given family of curves. $$y=e^{c x}$$

Determine the order of the differential equation. $$\left(\frac{d y}{d x}\right)^{3}+y^{2}=\sin x$$

Solve the given differential equation. $$(x-1)(x-2) y^{\prime \prime}=y^{\prime}-1$$

Use Euler's method with the specified step size to determine the solution to the given initial-value problem at the specified point. $$y^{\prime}=x-y^{2}, \quad y(0)=2, \quad h=0.05, \quad y(0.5)$$.

Determine whether the given differential equation is exact. $$\left(y+3 x^{2}\right) d x+x d y=0$$

Solve the given differential equation. $$x^{2} y^{\prime}-4 x y=x^{7} \sin x, \quad x > 0$$

A tank whose volume is \(200 \mathrm{L}\) is initially half full of a solution that contains \(100 \mathrm{g}\) of chemical. A solution containing 0.5 \(\mathrm{g} / \mathrm{L}\) of the same chemical flows into the tank at a rate of \(6 \mathrm{L} / \mathrm{min}\), and the well-stirred mixture flows out at a rate of 4 L/min. Determine the concentration of chemical in the tank just before the solution overflows.

A certain cell culture has a doubling time of 4 hours. Initially there were 2000 cells present. Assuming an exponential growth law, determine the time it takes for the culture to contain \(10^{6}\) cells.

Solve the given differential equation. $$e^{x+y} d y-d x=0$$

Determine the differential equation giving the slope of the tangent line at the point \((x, y)\) for the given family of curves. $$y=c x^{2}$$

Determine the order of the differential equation. $$y^{\prime \prime}+x y^{\prime}+e^{x} y=y^{\prime \prime \prime}$$

Solve the given differential equation. $$y^{\prime \prime}+2 y^{-1}\left(y^{\prime}\right)^{2}=y^{\prime}$$

Use Euler's method with the specified step size to determine the solution to the given initial-value problem at the specified point. $$y^{\prime}=-x^{2} y, \quad y(0)=1, \quad h=0.2, \quad y(1)$$.

Determine whether the given differential equation is exact. $$2 x e^{y} d x+\left(3 y^{2}+x^{2} e^{y}\right) d y=0$$

Solve the given differential equation. $$y^{\prime}+2 x y=2 x^{3}$$

Determine whether the given function is homogeneous of degree zero. Rewrite those that are as functions of the single variable \(V=y / x\). $$f(x, y)=\frac{\sqrt{3 x^{2}+5 y^{2}}}{2 x+5 y}, \quad x>0$$

A tank whose volume is 40 L initially contains 20 L of water. A solution containing \(10 \mathrm{g} / \mathrm{L}\) of salt is pumped into the tank at a rate of \(4 \mathrm{L} / \mathrm{min},\) and the well- stirred mixture flows out at a rate of 2 L/min. How much salt is in the tank just before the solution overflows?

At time \(t,\) the population \(P(t)\) of a certain city is increasing at a rate that is proportional to the number of residents in the city at that time. In January \(2000,\) the population of the city was 10,000 and by 2005 it had risen to 20,000. (a) What will the population of the city be at the beginning of the year \(2020 ?\) (b) In what year will the population reach one million?

Solve the given differential equation. $$\frac{d y}{d x}=\frac{y}{x \ln x}$$

Determine whether the differential equation is linear or nonlinear. $$\sin x \cdot y^{\prime \prime}+y^{\prime}-\tan y=\cos x$$.

Determine the differential equation giving the slope of the tangent line at the point \((x, y)\) for the given family of curves. $$y=c / x$$

Determine the order of the differential equation. $$\sin \left(y^{\prime \prime}\right)+x^{2} y^{\prime}+x y=\ln x$$

Use Euler's method with the specified step size to determine the solution to the given initial-value problem at the specified point. $$y^{\prime}=2 x y^{2}, \quad y(0)=0.5, \quad h=0.1, \quad y(1)$$.

Solve the given differential equation. $$\frac{d y}{d x}+\frac{2 x}{1-x^{2}} y=4 x, \quad-1 < x < 1$$

Determine whether the given function is homogeneous of degree zero. Rewrite those that are as functions of the single variable \(V=y / x\). $$f(x, y)=\frac{x+7}{2 y}$$

A tank initially contains \(20 \mathrm{L}\) of water. A solution containing \(1 \mathrm{g} / \mathrm{L}\) of chemical flows into the tank at a rate of \(3 \mathrm{L} / \mathrm{min},\) and the mixture flows out at a rate of 2 L/min. (a) Set up and solve the initial-value problem for \(A(t),\) the amount of chemical in the tank at time \(t\) (b) When does the concentration of chemical in the tank reach 0.5 g/L?

In the logistic population model \((1.5 .3),\) if \(P\left(t_{1}\right)=P_{1}\) and \(P\left(2 t_{1}\right)=P_{2},\) then it can be shown (through some tedious algebra to derive by hand, although easy on a computer algebra system) that $$\begin{aligned}r &=\frac{1}{t_{1}} \ln \left[\frac{P_{2}\left(P_{1}-P_{0}\right)}{P_{0}\left(P_{2}-P_{1}\right)}\right] \\\c=& \frac{P_{1}\left[P_{1}\left(P_{0}+P_{2}\right)-2 P_{0} P_{2}\right]}{P_{1}^{2}-P_{0} P_{2}}\end{aligned}$$ These formulas will be used. The initial population in a small village is \(500 .\) After5 years, this has grown to \(800,\) while after 10 years the population is \(1000 .\) Using the logistic population model, determine the population after 15 years.

Solve the given differential equation. $$y d x-(x-2) d y=0$$

Determine whether the differential equation is linear or nonlinear. $$\frac{d^{4} y}{d x^{4}}+3 \frac{d^{2} y}{d x^{2}}=x$$.

Determine the differential equation giving the slope of the tangent line at the point \((x, y)\) for the given family of curves. $$y^{2}=c x$$

Verify that, for \(t>0, y(t)=\ln t\) is a solution to the differential equation $$ 2\left(\frac{d y}{d t}\right)^{3}=\frac{d^{3} y}{d t^{3}} $$