Ma 341 section 1: Differential Equations

TYPOS, ERRORS, and more There are a few errors in the homework solutions and the notes, for you convenience I post the links again here,

Selected Homework Solutions:

These solutions may contain errors, if you think you found one send me an email with a clear exposition of what the error is and how to fix it and you'll earn a bonus point on some test, this also goes for the notes. Of course once I notify the class of the error you may no longer ask for that point. To start with check the list below. These are the corrections from previous classes, there are likely more to add, find a new one its worth a bonus point if you're the first to email the source and solution of the error:
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1.) Section 4.4 #36, you didn't multiply the 8 for the discriminant by "a" and "c," which would have given us sqrt(41) rather than sqrt(13) ==== sqrt(9+32) rather than sqrt(9+4)
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2.) 4.3 #12 you have r^2 + 7 = 0 then r = plus/minus sqrt(-7) then r = plus/minus 7i . just wondering where the sqrt went... so it should be r = plus/minus sqrt(7)i
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3.) 4.2#31 one of you pointed out that, and I quote, For what it's worth, on Section 4.2 #31 I think the explanation is right but also imperfect. You say that (tan t)^2 - (sec t)^2 = 1, but I think that actually equals (-1). *tan = sin/cos *sec = 1/cos ...so we have... (((sint)^2)/((cos t)^2)) - (1/((cos t)^2)) and that is over a common denominator, and.. ((sin t)^2) - 1 = -((cos t)^2) that gives us - ((cos t)^2)/((cos t)^2), or (-1) I think my notation could be much better, but you get the point.
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4.) 4.5 #22: another student pointed out, and I quote, I found an error in section 4.5 #22 of the homework. At the very end when you are calculating what the constant E should be you set it as 20 when it should be 2.
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5.) 4.3 #4: r=(10+-sqrroot(100-104))/2 so, r=5+-i therefore, y= exp(5x)(c1cos(x)+c2sin(x)) (I calculated the discriminant incorrectly.)
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6.) 2.2 #22: 1/3x^3 = -y^2 +4 (- in front of y was missing in the boxed answer)
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7.) 3.2 #2: I messed up the arithmetic simplifying (.3/.12) into 4/100 near the end. In fact .3/.12 = 30/12 = 5/2. this will change the answer.
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8.) 7.3# 6: I think that there is a problem with question #6 in homework 7.3 The Laplace Transform of exp(3t)*t^2 should be 2/(s-3)^3. You have it down as 2/(s-3)

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9.) 7.3# 12: One of you remarked, "First off you left off the "i" in sin(theta)=1/(2i).... Second, what is the i for anyways. It is there in all the sin and cos stuff and then it is just dropped in the last step. If that is just what it does, it doesn't seem like there is really any point to it. Thanks for your time" ******************************
To which I replied,
hmmm, the i=sqrt(-1), so i^2 = -1.
the point is that we can talk about exp(ix) if we want and it follows all the same rules as when we have exp(3x), but exp(ix) breaks up into a real and imaginary part,

exp(ix) = cos(x) +i*sin(x)

this is a basic fact ( well not basic in the sense of duh. everybody knows that, but basic in the sense that it can be taken as a definition)

This is a useful construction for many reasons,
1.) exp((a+ib)x) is a (complex) solution to a differential equation
which has a+ib as a solution to the aux. equation. Then since we
want real solutions we can take the Re{exp((a+ib)x)} or
Im{exp((a+ib)x)} which are exp(ax)cos(bx) and
exp(ax)sin(bx) respectively. A complex solution always encodes
two real solutions, just like a complex number has two real
numbers in it.

2.) exp(ix) = cos(x) +i*sin(x) can be used to calculate just about
any trig. identity under the sun. This is my main application
of complex variables tricks in chapter 7. And in particular 7.3#12
requires you be able to change the trig. functions you are given
into a sum of sines and cosines. The imaginary exponential will
allow you to derive any identity you might need to accomplish
this. You need to do this because otherwise the Laplace transform
is not known.

3.) there is a neat intepretation in terms of rotations in the plane.
But yes, bonus point for my missing i in the statement in the middle of the problem, fortunately I did include the i elsewhere so the overall solution is correct. Good catch though.
Thanks,

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10.) 7.4# 21: Your answer for 7.4 #21 is wrong. You put an extra "t" in each piece of the answer.

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11.) Example on solving system of ODEs via Laplace (on page 70-71 I think): Just a friendly reminder about your mistake on the example in the notes. You didn't multiply by the 1/2 at the top of pg. 71...leaving you with the random 2.

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12.) 7.5 #4: I think there is an error on 7.5 #4,
The third line should be
(s^2 + 6s +5)Y - s - 7 + 6 = 12/(s-1)
(there should not be a s after 6.)
So the partial fraction came out to be
Y=1/(s-1)-1/(s+1)-1/(s+5)
and
y = exp(t)-exp(-t)-exp(-5t)

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13.) 7.5# 5: Also, in section 7.5 #5, the second line of the solution should be:
s^2W - s (instead of +s) + 1+ W = 2/s^3 + 2/s
so the answer should be
w(t) = cos(t) - sin(t) + t^2
and that's what the answer on the textbook, too.

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14.) 9.4 #7: "In your solution, I believe you made an error.
You have z'=-w-t^2
Shouldn't the t^2 be positive here as well as in the final solution? That's the way it is in the book... "

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15.) 9.3 # 12: the solution to 9.3 # 12 is in fact the solution to 9.3 #11 ( I think its a good solution for that one, the book agrees with my answer)

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16.) 9.3 # 14: the solution to 9.3 # 14 is wrong because I copied the first row of the matrix in 14 yet then I proceeded to use the second and third rows from 16. That said I think the solution is correct for the matrix that I started with.

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17.) 9.6#13: There is a small problem with 9.6 #13 which does not effect the final answer. When you get to y= -2 + i, and describe it as a + iB, you mistakingly set B=2 instead of 1."

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18.) 2.3# 22: Just wanted to bring what I believe is an error to your attention. In section 2.3, question #22: after you found the integrating factor to be abs(sin(x)), you went back through and multiplied ONLY the RIGHT side of the equation by that integrating factor, leaving the left side simply as X as opposed to x*sin(x). Am I right or am I simply missing something else? I've checked and re-checked, just thought i'd let you know. Let me know if I'm correct. Thanks! ********************************************************************************
I responded with the following corrected solution,
Bonus Point. My posted solution is wrong, I dropped the sin(x) for some reason, well it did make the integration of the RHS a little easier. Instead we should have (assuming that x is such that |sin(x)| = sin(x) for now )
sin(x)dy/dx + cos(x)y = xsin(x)
Then the product rule in reverse gives,
d/dx (sin(x)y ) = xsin(x) (EQUATION BOB)
Recall that Int(xsin(x)dx) is done via integration by parts. Let u = x and
let dv = sin(x)dx so that du = dx and v = -cos(x), by IBP we have,
Int (xsin(x)dx) = uv - Int( vdu) = -xcos(x) + Int(cos(x)dx) = -xcos(x) + sin(x) + c
It is easy to verify that we did the integration correctly, just differentiate the answer and we get back xsin(x) as we ought. Returning to the EQUATION BOB integrate both sides to obtain,
sin(x)y = -xcos(x) + sin(x) + c
Thus,
y = -xcos(x)/sin(x) + 1 + c/sin(x)
the case |sin(x)| = -sin(x) can be dealt with by a similar calculation.
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19.)In looking at the Selected HW Solutions I believe I came upon an error. Section 1.2 problem # 1 part b asks to show that a function solves the differential equation. In the 6th line of the solution it says “y’’ + y squared = …” I believe the y’’ should be y’, since just above we solved for y’ and then squared it to create the 6th line’s equation by adding the two together. If this is not correct, please let me know, since it’s the only way it makes sense to me.

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I responded,
You are correct. The y'' is supposed to be y'. Bonus point awarded.

20.) Not a terribly critical question, but in doing Test 1 from September 18, 2007 problem 8, at the bottom of the page when solving for A, B, and C, I got B = -2 and A = 2 from the exact same formulas. Because C=1, B must be -2 for B+2C=0, and A must be 2 for A+B=0.

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I responded,
You are correct, my last couple of algebra steps were flawed. Thanks. Bonus point.

21.) E5 on page 120 of my notes has exp(-2t) type terms when I meant for those to read exp(2t) since the eigenvalue was two.

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Last Modified: 6-11-08