Chapter 7

Find and interpret all equilibrium points for the predator-prey model. $$\left\\{\begin{array}{l}x^{\prime}=0.2 x-0.2 x^{2}-0.4 x y \\\ y^{\prime}=-0.1 y+0.2 x y\end{array}\right.$$

Determine whether the differential equation is separable. $$y^{\prime}=(3 x+1) \cos y$$

Find the solution of the given differential equation satisfying the indicated initial condition. $$y^{\prime}=4 y, y(0)=2$$

Construct four of the direction field arrows by hand and use your CAS or calculator to do the rest. Describe the general pattern of solutions. $$y^{\prime}=\sqrt{x^{2}+y^{2}}$$

Find and interpret all equilibrium points for the predator-prey model. $$\left\\{\begin{array}{l}x^{\prime}=0.4 x-0.1 x^{2}-0.2 x y \\\ y^{\prime}=-0.2 y+0.1 x y\end{array}\right.$$

Determine whether the differential equation is separable. $$y^{\prime}=2 x(\cos y-1)$$

Find the solution of the given differential equation satisfying the indicated initial condition. $$y^{\prime}=3 y, y(0)=-2$$

Find and interpret all equilibrium points for the predator-prey model. $$\left\\{\begin{array}{l}x^{\prime}=0.3 x-0.1 x^{2}-0.2 x y \\\ y^{\prime}=-0.2 y+0.1 x y\end{array}\right.$$

Determine whether the differential equation is separable. $$y^{\prime}=(3 x+y) \cos y$$

Find the solution of the given differential equation satisfying the indicated initial condition. $$y^{\prime}=-3 y, y(0)=5$$

Find and interpret all equilibrium points for the predator-prey model. $$\left\\{\begin{array}{l}x^{\prime}=0.1 x-0.1 x^{2}-0.4 x y \\\ y^{\prime}=-0.1 y+0.2 x y\end{array}\right.$$

Determine whether the differential equation is separable. $$y^{\prime}=2 x(y-x)$$

Find the solution of the given differential equation satisfying the indicated initial condition. $$y^{\prime}=-2 y, y(0)=-6$$

Find and interpret all equilibrium points for the predator-prey model. $$\left\\{\begin{array}{l}x^{\prime}=0.2 x-0.1 x^{2}-0.4 x y \\\ y^{\prime}=-0.3 y+0.1 x y\end{array}\right.$$

Determine whether the differential equation is separable. $$y^{\prime}=x^{2} y+y \cos x$$

Find the solution of the given differential equation satisfying the indicated initial condition. $$y^{\prime}=2 y, y(1)=2$$

Find and interpret all equilibrium points for the predator-prey model. $$\left\\{\begin{array}{l}x^{\prime}=0.2 x-0.1 x^{2}-0.4 x y \\\ y^{\prime}=-0.2 y+0.1 x y\end{array}\right.$$

Determine whether the differential equation is separable. $$y^{\prime}=2 x \cos y-x y^{3}$$

Find the solution of the given differential equation satisfying the indicated initial condition. $$y^{\prime}=-y, y(1)=2$$

Determine whether the differential equation is separable. $$y^{\prime}=x^{2} y-x \cos y$$

Find the solution of the given differential equation satisfying the indicated initial condition. $$y^{\prime}=-3, y(0)=3$$

Determine whether the differential equation is separable. $$y^{\prime}=x^{3}-2 x+1$$

Find the solution of the given differential equation satisfying the indicated initial condition. $$y^{\prime}=-2, y(0)=-8$$

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

Involve exponential growth. Suppose a bacterial culture doubles in population every 4 hours. If the population is initially \(100,\) find an equation for the population at any time. Determine when the population will reach 6000

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

Involve exponential growth. Suppose a bacterial culture triples in population every 5 hours. If the population is initially \(200,\) find an equation for the population at any time. Determine when the population will reach 20,000

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

Involve exponential growth. Suppose a bacterial culture initially has 400 cells. After 1 hour, the population has increased to \(800 .\) Find an equation for the population at any time. What will the population be after 10 hours?

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

Involve exponential growth. Suppose a bacterial culture initially has 100 cells. After 2 hours, the population has increased to \(400 .\) Find an equation for the population at any time. What will the population be after 8 hours?

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. $$y^{\prime}=2 x y, y(0)=1$$

The differential equation is separable. Find the general solution, in an explicit form if possible. Sketch several members of the family of solutions. $$y^{\prime}=\frac{6 x^{2}}{y\left(1+x^{3}\right)}$$

Involve exponential growth. A bacterial culture grows exponentially with growth constant 0.12 hour \(^{-1} .\) Find its doubling time.

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. $$y^{\prime}=x / y, y(0)=2$$

The differential equation is separable. Find the general solution, in an explicit form if possible. Sketch several members of the family of solutions. $$y^{\prime}=\frac{3 x}{y+1}$$

Involve exponential growth. A bacterial culture grows exponentially with growth constant 0.12 hour \(^{-1} .\) Find its doubling time.

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. $$y^{\prime}=4 y-y^{2}, y(0)=1$$

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

Involve exponential growth. Suppose that a population of \(E .\) coli doubles every 20 minutes. A treatment of the infection removes \(90 \%\) of the \(E .\) coli present and is timed to accomplish the following. The population starts at size \(10^{8},\) grows for \(T\) minutes, the treatment is applied and the population returns to size \(10^{8} .\) Find the time \(T\)

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. $$y^{\prime}=x / y^{2}, y(0)=2$$

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

Involve exponential growth. Research by Meadows, Meadows, Randers and Behrens indicates that the earth has \(3.2 \times 10^{9}\) acres of arable land available. The world population of 1950 required \(10^{9}\) acres to sustain it, and the population of 1980 required \(2 \times 10^{9}\) acres. If the required acreage grows at a constant percentage rate, in what year will the population reach the maximum sustainable size?

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. $$y^{\prime}=1-y+e^{-x}, y(0)=3$$

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

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

Suppose some quantity is increasing exponentially (e.g., the number of cells in a bacterial culture) with growth rate \(r .\) Show that the doubling time is \(\frac{\ln 2}{r}\)

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. $$y^{\prime}=\sin y-x^{2}, y(0)=1$$

Find and interpret all equilibrium points for the competing species model. (Hint: There are four equilibrium points in exercise 17. $$\left\\{\begin{array}{l} x^{\prime}=0.4 x-0.1 x^{2}-0.2 x y \\ y^{\prime}=0.5 y-0.4 y^{2}-0.1 x y \end{array}\right.$$

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