MAT 303: Homework 3
due Monday, September 26

In these exercises, we want to use Mathematica to visualize solutions of first order differential equations and gain further insight into their long term behavior. Please read the notes posted on the Course Webpage before solving these exercises.

Problem 3.1

Study the differential equation (dy/dx)=((4x-(x^3))/(4+(y^3))) using Mathematica.

Plot the vector field using VectorPlot[...] in the region x ∈ [-3.5, 3.5], y ∈ [-3.3, 3.3]. Use an option of VectorPlot[...] that plots some typical integral curves on top of the vector field.

Plot the vector field using StreamPlot[...] in the same region x ∈ [-3.5, 3.5], y ∈ [-3.3, 3.3]. On the same plot, highlight the solution curve passing through the point (0, 1) in Red, and the solution (integral curve) passing through the point (1, -1.5) in Green.

Explain mathematically the discontinuities of the vector field (where the direction of the arrows changes, and the slopes of the integral curves are ∞ or -∞). Then find numerically the places where these discontinuities occur using NSolve[expr,vars]. This command attempts to find numerical approximations to the solutions of the system expr of equations or inequalities for the variables vars.

The example below shows an application of NSolve (notice the double equal sign == that is used to test equality between expressions).

NSolve[x^2+3x^3-x^4==0, x]

The ODE given in Problem 3.1 is a separable differential equation, that we can solve and get the family of solutions defined by the implicit equation  4y+(1/4)(y^4)=2(x^2)-(1/4)(x^4)+c . Plot together on the same picture the implicit solutions corresponding to the following choices of the constant c=0,1,2,-1,-4,-5. Is the picture consistent to the Vector Field pictures obtained in part 1. and part 2.?

The function ContourPlot[expr, vars] can be used to plot the solutions of an implicit equation.  The examples below show an application of ContourPlot.

ContourPlot[{x^2+y^2==1/4},{x,-1,1},{y,-1,1}]

ContourPlot[{x^2+y^2==1/4,(x^2+y^2)^2==x^2-y^2},{x,-1,1},{y,-1,1},ContourStyle->{GrayLevel[0],Dashing[{.03}]}]

Problem 3.2

Study the differential equation (dy/dx)=x-y using analytical methods together with Mathematica.

Plot the vector field using VectorPlot[...] or StreamPlot[...] in the region x ∈ [-5, 5], y ∈ [-5, 5]. On the same vector field, draw the integral curve corresponding to the initial condition (2,1). What do you notice?

Solve the differential equation in two ways:

by the Integrating Factor Method, noting the fact that this is a linear equation.

by the Substitution Method, by doing the substitution v = x-y.

Define a function in Mathematica, called sol[x_,c_] that outputs the solution that you have computed at part 2. Use the function Plot[...] to plot togther several solutions of the differential equation.

Explain the presence of a linear solution y(x)=x-1 of the differential equation. Is this a singular solution?

Show that every solution curve approaches this straight line y(x)=x-1 as x→+∞. Is this asymptotic behavior of solutions predicted by the vector field that you have plotted in part 1. ?

Analyze the local and global existence and uniqueness of solutions of the differential equation given in the problem. Is the straight line y(x)=x-1 an obstruction to global uniqueness of solutions (in other words, can it be used to produce infinitely many solutions)?