Differential Equations: Math 3226

Problem 1. Solve

$$y' + \cot(x) y = \sin(2x),\;\;\; y\left(\frac{\pi}{2}\right) = 1.$$

Problem 2. Consider the initial-value problem

$$\frac{dy}{dt} = y^2\;,\;\;y(0)=1$$

Using Euler's method,compute three different approximate solutions corresponding to $\Delta t = 1$, $\Delta t = 0.5$, and $\Delta t = 0.25$ over the interval $0 \leq t \leq 4$. Graph all three solutions. What predictions do you make about the actual solution to the initial-value problem? How do the graphs of these approximate solutions relate to the graph of the actual solution? Why?

Problem 3. The graph of f(y) is given.

Sketch the phase line and some solutions for the differential equation dy/dt=f(y).

Problem 4. Consider the system

$$\left\{\begin{array}{lcl} \displaystyle \frac{dx}{dt} &=& y-y... ... &&\\ \displaystyle \frac{dy}{dt} &=& x\\ \end{array}\right.$$

1.

Sketch the nullclines and find the direction of the vector field along the nullclines.

2.

Show that there is at least one solution in each of the second and fourth quadrants that tends to the origin as $t \rightarrow +\infty$. Similarly, show that there is at least one solution in each of the first and third quadrants that tends to the origin as $t \rightarrow -\infty$.

3.

Describe the behavior of solutions near the equilibrium points.

Problem 5. Consider the system

$$\left\{\begin{array}{lcl} \displaystyle \frac{dx}{dt} &=& -x ... ...&\\ \displaystyle \frac{dy}{dt} &=& ax-y\\ \end{array}\right.$$

1.

Find the value of a which gives a system with repeated or double eigenvalues.

2.

Consider the system with the value of a found in 1. Find the general solution.

3.

Find the particular solution which satisfies

Y₀ = (1,1)

Problem 6. Consider the harmonic oscillator with Mass m=1, spring constant k_s = 1, and coefficient of damping k_d.

1.

Write the corresponding second order equation and first order system,

2.

Determine all values of k_d at which a bifurcation occurs; and

3.

Give the general solution of the system when k_d = 1,and k_d=2.

Problem 7. Consider the system

$$\left\{\begin{array}{lcl} \displaystyle \frac{dx}{dt} &=& -x ... ...&\\ \displaystyle \frac{dy}{dt} &=& x-2y\\ \end{array}\right.$$

1.

Find the equilibrium points.

2.

Classify the equilibrium points as: source, sink, center, and so on...

3.

Sketch the phase-plane near the point (0,0).

Problem 8. Find the solution to the initial value problem

$$y'' + y = g(t),\;\; y(0) = 1,\;\;y'(0) = 0,$$

where

$$g(t) = \left\{ \begin{array}{lll} \sin(2t),& \;\;\;\mbox{if}... .../2\\ 0,& \;\;\; \mbox{if}\;\; \pi/2 < t \end{array} \right.$$

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