Problem 1
For the differential equations in Exercises 1-4, perform each of the following tasks. a) Print out the direction field for the differential equation with the display window defined by \(t \in[-5,5]\) and \(y \in[-5,5]\). You might consider increasing the number of field points to 25 in the DFIELD6 Window settings dialog box. On this printout, sketch with a pencil as best you can the solution curves through the initial points \(\left(t_{0}, y_{0}\right)=(0,0),(-2,0),(-3,0),(0,1)\), and \((4,0)\). Remember that the solution curves must be tangent to the direction lines at each point. b) Use dfield6 to plot the same solution curves to check your accuracy. Turn in both versions. $$ y^{\prime}=t y . $$
Problem 2
For the differential equations in Exercises 1-4, perform each of the following tasks. a) Print out the direction field for the differential equation with the display window defined by \(t \in[-5,5]\) and \(y \in[-5,5]\). You might consider increasing the number of field points to 25 in the DFIELD6 Window settings dialog box. On this printout, sketch with a pencil as best you can the solution curves through the initial points \(\left(t_{0}, y_{0}\right)=(0,0),(-2,0),(-3,0),(0,1)\), and \((4,0)\). Remember that the solution curves must be tangent to the direction lines at each point. b) Use dfield6 to plot the same solution curves to check your accuracy. Turn in both versions. $$ y^{\prime}=y^{2}-t^{2} $$
Problem 3
For the differential equations in Exercises 1-4, perform each of the following tasks. a) Print out the direction field for the differential equation with the display window defined by \(t \in[-5,5]\) and \(y \in[-5,5]\). You might consider increasing the number of field points to 25 in the DFIELD6 Window settings dialog box. On this printout, sketch with a pencil as best you can the solution curves through the initial points \(\left(t_{0}, y_{0}\right)=(0,0),(-2,0),(-3,0),(0,1)\), and \((4,0)\). Remember that the solution curves must be tangent to the direction lines at each point. b) Use dfield6 to plot the same solution curves to check your accuracy. Turn in both versions. $$ y^{\prime}=2 t y /\left(1+y^{2}\right) $$
Problem 4
For the differential equations in Exercises 1-4, perform each of the following tasks. a) Print out the direction field for the differential equation with the display window defined by \(t \in[-5,5]\) and \(y \in[-5,5]\). You might consider increasing the number of field points to 25 in the DFIELD6 Window settings dialog box. On this printout, sketch with a pencil as best you can the solution curves through the initial points \(\left(t_{0}, y_{0}\right)=(0,0),(-2,0),(-3,0),(0,1)\), and \((4,0)\). Remember that the solution curves must be tangent to the direction lines at each point. b) Use dfield6 to plot the same solution curves to check your accuracy. Turn in both versions. $$ y^{\prime}=y(2+y)(2-y) $$
Problem 5
Use dfield6 to plot a few solution curves to the equation \(x^{\prime}+x \sin (t)=\cos (t)\). Use the display window defined by \(x \in(-10,10)\) and \(t \in(-10,10)\).
Problem 6
Use dfield6 to plot the solution curves for the equation \(x^{\prime}=1-t^{2}+\sin (t x)\) with initial values \(x=\) \(-3,-2,-1,0,1,2,3\) at \(t=0\). Find a display window which shows the most important features of the solutions by experimentation.
Problem 7
For the differential equations in Exercises \(7-10\) perform the following tasks. a) Use dfield6 to plot a few solutions with different initial points. Start with the display window bounded by \(0 \leq t \leq 10\) and \(-5 \leq y \leq 5\), and modify it to suit the problem. Print out the display window and turn it in as part of this assignment. b) Make a conjecture about the limiting behavior of the solutions of as \(t \rightarrow \infty\). c) Find the general analytic solution to this equation. d) Verify the conjecture you made in part b), or if you no longer believe it, make a new conjecture and verify that. $$ y^{\prime}+4 y=8 $$
Problem 8
For the differential equations in Exercises \(7-10\) perform the following tasks. a) Use dfield6 to plot a few solutions with different initial points. Start with the display window bounded by \(0 \leq t \leq 10\) and \(-5 \leq y \leq 5\), and modify it to suit the problem. Print out the display window and turn it in as part of this assignment. b) Make a conjecture about the limiting behavior of the solutions of as \(t \rightarrow \infty\). c) Find the general analytic solution to this equation. d) Verify the conjecture you made in part b), or if you no longer believe it, make a new conjecture and verify that. $$ \left(1+t^{2}\right) y^{\prime}+4 t y=t $$
Problem 11
In Exercises 11-14 we will consider a certain lake which has a volume of \(V=100 \mathrm{~km}^{3}\). It is fed by a river at a rate of \(r_{i} \mathrm{~km}^{3} / \mathrm{year}\), and there is another river which is fed by the lake at a rate which keeps the volume of the lake constant. In addition, there is a factory on the lake which introduces a pollutant into the lake at the rate of \(p \mathrm{~km}^{3} / \mathrm{year}\). This means that the rate of flow from the lake into the outlet river is \(\left(p+r_{i}\right) \mathrm{km}^{3} / \mathrm{year}\). Let \(x(t)\) denote the volume of the pollutant in the lake at time \(t\), and let \(c(t)=x(t) / V\) denote the concentration of the pollutant. Show that, under the assumption of immediate and perfect mixing of the pollutant into the lake water, the concentration satisfies the differential equation \(c^{\prime}+\left(\left(p+r_{i}\right) / V\right) c=p / V\).
Problem 14
Rivers do not flow at the same rate the year around. They tend to be full in the Spring when the snow melts, and to flow more slowly in the Fall. To take this into account, suppose the flow of our river is $$ r_{i}=50+20 \cos (2 \pi(t-1 / 3)) $$ Our river flows at its maximum rate one-third into the year, i.e., around the first of April, and at its minimum around the first of October. a) Setting \(p=2\), and using this flow rate, use dfield6 to plot the concentration for several choices of initial concentration between \(0 \%\) and \(4 \%\). (If your solution seems erratic, reduce the relative error tolerance using \(\mathbf{O p t i o n s} \rightarrow \mathbf{S o l v e r}\) settings.) How would you describe the behavior of the concentration for large values of time? b) It might be expected that after settling into a steady state, the concentration would be greatest when the flow was smallest, around the first of October. At what time of the year does the highest concentration actually occur? Reduce the error tolerance until you get a solution curve smooth enough to make an estimate.
