MAT244H1 Study Guide - Midterm Guide: Jordan Bell, Wronskian, Stationary Point
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Mat 244 Practice Test 3
July 25, 2013
You should be able to do all of the questions in this test given enough time.
This is certainly longer than the actual test will be. I wanted to give you more
worked out examples for calculus of variations.
1. Find the extremals of the functional
Solution. f=y2+y02−2ysin x. So
∂y = 2y−2 sin x, ∂f
Hence the Euler-Lagrange equation is
2y−2 sin x= 2y00
y00 −y=−sin x.
This is a second order ODE. The solution of the homogeneous equation is
Now we use variation of parameters. y1=ex, y2=e−x. The Wronskian is
W=−2. g=−sin x.
To compute u1, u2we need to use integration by parts. I won’t write this out;
it is something that you should be able to skillfully do if you have suﬃcient time,
and you should have absolutely no problem remembering how to do integration
by parts. You don’t have to do it in your head; unless it’s obvious to me, I
manually set uand dv and write out the integration by parts formula.
4e−x(cos x+ sin x)
4ex(−cos x+ sin x)
So the extremals of the functional Rx2
x1y2+y02−2ysin xdx are
4(cos x+ sin x) + 1
4(−cos x+ sin x)
2. Find the extremals of the functional
I(y) = Zx2
This takes longer than I would want for a test question, but if I gave you
enough time I would expect you to be able to do something like this level of
y02. Since fdoesn’t depend on x, the Euler-Lagrange
equation for this functional is
which can also be written ∂f
∂y0=−2(1 + y2)
Hence 2(1 + y2)
y02+1 + y2
1 + y2=c1y02
so (in this step I change c1, which is logically acceptable untl we have introduced
y0=c1p1 + y2
c1p1 + y2=dx.