91Ó°ÊÓ

Use a computer algebra system to (a) graph the slope field for the differential equation and (b) graph the solution satisfying the specified initial condition. $$ \frac{d y}{d x}=\frac{1}{2} e^{-x / 8} \sin \frac{\pi y}{4}, \quad y(0)=2 $$

The value of a tract of timber is \(V(t)=100,000 e^{0.8 \sqrt{t}}\) where \(t\) is the time in years, with \(t=0\) corresponding to 1998 . If money earns interest continuously at \(10 \%\), the present value of the timber at any time \(t\) is \(A(t)=V(t) e^{-0.10 t}\). Find the year in which the timber should be harvested to maximize the present value function.

The logistic equation models the growth of a population. Use the equation to (a) find the value of \(k\), (b) find the carrying capacity, (c) find the initial population, (d) determine when the population will reach \(\mathbf{5 0 \%}\) of its carrying capacity, and (e) write a logistic differential equation that has the solution \(P(t)\). \(P(t)=\frac{1500}{1+24 e^{-0.75 t}}\)

Use Euler's Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use \(n\) steps of size \(h\). $$ y^{\prime}=3 x-2 y, \quad y(0)=3, \quad n=10, \quad h=0.05 $$

The logistic equation models the growth of a population. Use the equation to (a) find the value of \(k\), (b) find the carrying capacity, (c) find the initial population, (d) determine when the population will reach \(\mathbf{5 0 \%}\) of its carrying capacity, and (e) write a logistic differential equation that has the solution \(P(t)\). \(P(t)=\frac{5000}{1+39 e^{-0.2 t}}\)

The logistic differential equation models the growth rate of a population. Use the equation to (a) find the value of \(k\), (b) find the carrying capacity, (c) use a computer algebra system to graph a slope field, and (d) determine the value of \(P\) at which the population growth rate is the greatest. \(\frac{d P}{d t}=3 P\left(1-\frac{P}{100}\right)\)

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. In linear growth, the rate of growth is constant.

Use Euler's Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use \(n\) steps of size \(h\). $$ y^{\prime}=\cos x+\sin y, \quad y(0)=5, \quad n=10, \quad h=0.1 $$

Explain how to interpret a slope field.

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(y=f(x)\) is a solution of a first-order differential equation, then \(y=f(x)+C\) is also a solution.