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1

Term Test 2 / TT2 Question 2

« on: March 27, 2013, 10:02:33 PM »

Consider the second order equation
\begin{equation*}
x''=x^4-5x^2+4
\end{equation*}


(a) Reduce to the first order system in variables $(x, y, t)$  with $y = x'$, i.e.
\begin{equation*}
\left\{ \begin{array}{ll}
x'=\ldots\\
y'=\ldots\\
\end{array}\right.
\end{equation*}

(b) Find solution in the form $H(x,y)=C$.

(c) Find critical points and linearize system in these points.

(d)  Classify the linearizations at the critical points (i.e. specify  whether they are nodes, saddles, etc., indicate stability and, if applicable,  orientation) and sketch their phase portraits.

(e) Sketch the phase portraits of the nonlinear system near each of  the critical points.

(f) Sketch the solutions on $(x,y)$ plane.

2

Ch 7 / Chapter 7.9: Laplace Transforms

« on: March 25, 2013, 02:24:45 PM »

Are we expected to know how to use a Laplace transform to solve a non-homogeneous system?
This material is covered in chapter 6, which I do not know if we will cover by the end of the year. I cannot think of a type of system where a solution can only be obtained from this method, so I'm hesitant to learn it if we will always be allowed to pick which method to use when solving a non-homogeneous system.
More generally my question is, even if we do not cover it in class, how marginal will the value of this method be compared to undetermined coefficients or variation of parameters, on the final or future courses?