Ma 341 section 4 Homepage
Welcome, please note that the schedule and syllabus are linked just below.
Useful Materials and Links:
Course Syllabus for ma 341-004 ( grading scheme office hours etc...)
Course Schedule ( suggested problems assigned and test dates )
Differential Equations covered in calculus II A synopsis of things you should have seen before. We will rehash much of the subject matter from Ma 241 as we cover the more general case in this course (n=2 or 1 in Ma 241, we cover n=1,2,3,4,...).
Solutions to Tests this semester:
Test I (2007) Solution
Test II (2007) Solution to in-class part
Test III (2007) Solution
Old Tests and Solutions:
Test I (2006)
Test I (2006) Solution
Test II (2006) and Solution
Test III (2006) and Solution
Lecture will often closely follow these notes.
Introduction We define the basic vocabulary and discuss direction fields, Euler's Method and separation of variables. A table of contents links these notes to the corresponding sections in your text.
First Order ODEs We introduce the integrating factor method to solve tricky 1st order ODEs that separation of variables cannot handle then define exact equations and how to solve those.
applications Mixing tank problems, Newton's Law of Cooling, and Newtonian Mechanics. All phrase problem as a differential equation.
n-th order homogeneous ODE Theory of linear ODEs, linear independence, the Wronskian, differential and linear operators all used to motivate the solution to the general n-th order linear ODE with constant coefficients.
n-th order nonhomogenous ODE Extending our knowledge from the homogeneous case we learn two main methods to find particular solutions. The metod of undetermined coefficients asks us to "guess" a particular form for the particular solution which we then must determine the coefficients A,B,C,... by doing some algebra. The annihilator method explains those "guesses". Variation of parameters requires integration in contrast to undetermined coefficients and as such it is a weapon of last resort, it can of course treat more difficult examples at the cost of some hard integrations.
Laplace Transforms Basic definitions of the Laplace transform given. Then various theorems discussed that allow us to transform a differential equation in "t" to a corresponding algebraic equation in "s".
Inverse Laplace Transforms We learn how to reverse directions. Given an algebraic expression in "s" we find a corresponding expression in "t" ( not usually differential since we solved the algebra )
Discontinous functions and Laplace Transformations Here lies the real reason to discuss Laplace transformations. We can deal with discontinuous forcing functions in an easy manner that doesn't require us to break up into cases (like we would have to otherwise )
Convolution An interesting concept which gives us another tool for performing inverse Laplace transforms.
Dirac Delta Function Probably the weirdest thing in the whole course, you'll see.
Laplace Transformations and Systems of ODEs We follow same idea as in the case of just one equation, difference is that instead of getting a single algebraic equation to solve whereas for a system of differential equations we get a system of algebraic equations to solve. Then of course we have to take the inverse transform to finish the job.
Two Equations, Two Unknowns Systems of equations can be surprising at times, but it all comes back to the cases that arise in this simple case. The methods presented in this section are useful for many problems in this course and elsewhere.
what is the matrix ? Far from being a surreal expidition into philosophy-laden science fiction, the matrix is a notational convenience which allows us to solve many equations as if they were one single equation. The methods presented here are useful in many many many places.
Theory for systems We see how systems of 1st order linear ODEs have a special position in the big-picture of things, any linear n-th order linear ODE can be viewed as a system of 1st order ODEs. On the flipside we can expect certain features from our experience with the n-th order case, indeed much of the theory is analogus.
Simple Homogeneous Linear systems I say "Simple" because we examine only the cases where there are enough eigenvectors to comprise the solution. Interestingly, we can have a double root and still no factor of "t" appear since the presence of vectors in the solution introduces a new facet to the linear independence of our solutions.
Nonhomogeneous Linear systems Again we discuss two methods to extract the particular solution for a nonhomogeneous linear system of ODEs. Undetermined coefficients is considerably more difficult in this context, however variation of parameters inherits an elegance from the matrix formalism which makes it more tractable in my limited experience.
Matrix Exponential Here we learn how to solve the less than "Simple" cases. It is the case that exp(tA) is a fundamental matrix for the system of ODEs x'=Ax, but what does that mean? And how do we even calculate such a thing? We introduce the "generalized eigenvector" to aid us in this task, in the process we gain greater insight into why the eigenvector methods worked before. Eigenvectors are also useful in many applications besides the one under consideration in these notes, so experience in calculating eigenvectors is not a bad thing.
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.
Homework on prerequisite materials and generalizations thereof.
Corrections upto the first test:
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)
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
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.
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.
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.)
6.) 2.2 #22: 1/3x^3 = -y^2 +4
(- in front of y was missing in the boxed answer)
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.
Homework on Laplace Transforms
Homework on systems of differential equations
Bonus Point: First person to email me the identity of the scientist pictured below:
Hannah says do your homework early and ask good questions. You can't argue with the Hannah, she's always right.
Wild world of maple sheets:
I have gathered together some simple applications of Maple to aid you in completing your homework.Maple is not required for this course,but it might be helpful to use to check your answers (I also recommend the TI-89 as a supplement to brute force calculation, I'll allow you to use the calculator to check your work, but not to do the work.)
Integration indefinite and definite integrals.
Partial Fractions how to check your algebra with Maple
Direction Fields pretty Maple plots of DEqns
Solving ODEs Maple finds general solution for ODEqns
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Last Modified: 6-21-07