Math 2C03 - McMaster University

Chapter 10

2010-03-31T12:21:00.000-07:00

The past four lectures were covering chapter 10, and we covered sections 10.1, 10.2, touched on 10.3 and 10.3, and we covered 10.7. Specifically, we first started with the wave equation (section 10.7), where we used the method of separation of variables to reduce the problem to a boundary value problem. We then solved the boundary value problem (section 10.1), and then we introduced Fourier series (section 10.2-10.4) to prove the wave equation completely. We have not talked about the Heat Equation \[ \alpha^2 u_{xx}=u_t, \]
(sections 10.5), but the method for solving this is exactly the same as the wave equation, although the results are a bit different. We also covered a simpler version of theorem 10.3.1 (Fourier convergence theorem), and we didn't talk about what happens at the points of discontinuity.

Next assignment

2010-03-13T17:14:00.001-08:00

Next assignment is up, and this one has a hand in part. Try to write up complete solutions to the problem number 5, which is essentially a selection of problems 22-29 of chapter 5.3.

Laplace transform

2010-02-24T08:03:00.000-08:00

The past two classes we have introduced the concept of Laplace transform. The basic definition is $$ F[s]=\mathcal{L}[f]=\int_0^\intfy f(t)e^{-st}dt. $$ That is, given a function $f$, a function of $t$, $\mathcal{L}[f]$ spits out a function of s. This definition is very useful, and for the next little while, we will use it many times. Some of the properties of $ \mathcal{L} $ are the following:

$ \mathcal{L} $ is linear
If $f$ satisfies $|f(t)| < K e^{at}$ for some $a$ and $K$, then $ \mathcal{L}[f] $ converges for $s>a$.
$ \mathcal{L}[f^\prime] = s\mathcal{L}[f]-f(0)$
$ \mathcal{L}[e^{at}] = {1 \over s-a}.$

Page 319 of the book has a table of the Laplace transform of some of the standard functions. Example done in class, along with the examples in chapter 6.2 show how this transform can reduce a differential equation into simple algebraic expressions, and that reduces the provlem of solving a differential equation to finding the inverse Laplace transform of an algebraic expression. Example 1 and 2 in chapter 6.2 show how this can be done in simple cases.

Problems in pages 312-314 are great practice for getting comfortable with Laplace transform. Problems 1-4 are very important! If you are doubtful about them, you should come and talk to me about them as soon as possible. Problem 5-20 fill in some parts of the table on page 319, and are good practice for you. Problems 26 and 27 are some of the more tricky entries in page 319, which are within your ability to do. Problems in pages 322-325 are practice for calculating inverse Laplace transform and solving linear ODEs using laplace transform. Problems 1-10 are basic inverse transform, while problems 11-23 are for solving linear ODEs (as it was mentioned in class, for these problems using methods of chapter 3 and 4 is probably easier). Problem 24-26 are very important again, since methods of chapter 3 and 5 can not solve them. Problems 27-34 shows how using a Taylor series expansion you can calculate some of Laplace transforms. This is in turn is used to solve problem 35 and 36, which are very difficult differential equations. Problems 37 and 38 are examples of problems that you can solve using methods that we've talked about in class today.

midterm on Tuesday,

2010-02-19T17:08:00.000-08:00

Quick reminder that your midterm is on February 23rd at 3:30. Midterm is happening in room T28.

There are few of you that have a conflict on that time slot. If that's the case, you should email me about it. The alternate time for those people is at 4:30 in room UH/112.

Also, I've posted my answers to the sample questions on the midterm on my webpage.

Also, I've posted this in the comment section earlier (probably not the best spot), but here is a quick outline on what is covered in the midterm:
Everything we have covered in the class so far is fair game for the midterm. That includes basic definition of what is a differential equation (chapter 1), linear first order equations (2.1), separable equations (2.2), modelling using DE (2.3), a bit of existence uniqueness (2.4, although we mostly covered the uniqueness part), autonomous equations (2.5), exact equations (2.6), Euler's method (2.7), and all of higher order linear differential equations (chapters 3 and 4) except for the last section of chapter 3 on forced vibration and resonance frequency.

As such, the sample midterm, all of midterm 1, and problems 2 and 3 on second midterm are relevant for your studies.

Best of luck,
Soroosh

Non homogeneous equations

2010-02-09T11:48:00.000-08:00

It's been a while since I made a post, and during that time we've covered sections 3.4-3.7, and all of chapter 4. Today we've covered 3.7 and 4.4.

For this post, I will assume that you are now comfortable with materials in chapter 3.4, 3.5, 4.1, and 4.2, that is given any homogeneous linear equation with constant coefficient, you can find a general solution to this equation, and given any such innitial value problem, you can find a specific solution.

On Friday, we started looking at non-homogeneous linear differential equations (chapter 3.6): $L[y]=g(t)$, where $L[y]=a_ny^(n)+\cdots+a_1y'+a_0y$. In that case, we noted that if $g(t)$ is of the form $P(t)e^{at}\cos(bt)+Q(t)e^{at}\sin(bt)$ for some polynomial $P$ and $Q$, then we can find a particular solution of the form $y=((A_nt^n+\cdots+A_1t+a_0)e^{at}\cos(bt)+(B_nt^n+\cdots+B_1t+B_0)e^{at}\sin(bt))t^s$, and in fact we presented a general way of finding general solution of such cases (Page 180-181.)

On Monday, we talked about the method of variation of parameters. Here when solving $L[y]=g(t)$, we first find the general solution to $L[y_h]=0$, say $y_h=c_1y_1+\cdots c_ny_n$. Then we subsitute $L[c_1(t)y_1+\cdots c_n(t)y_n]=g(t)$ to get a linear systems of equations in $c_1,c_2,...,c_n$. We then make the assumption that
\[ \array{ c_1^\prime y_1+c_2^\prime y_2+\cdots + c_n^\prime y_n &=& 0 \\ c_1^\prime y_1^\prime +c_2^\prime y_2^\prime +\cdots+c_n^\prime y_n^\prime &=& 0 \\ \ldots &\ldots& \ldots \\ c_1^\prime y_1^{(n-2)} +c_2^\prime y_2^{(n-2)}+\cdots c_n^\prime y_n^{(n-2)} &=& 0. } \]
Substituting these assumption into $L[y]=g(t)$ we get \[ c_1^\prime y_1^{(n-1)}+c_2^\prime y_2^{(n-1)}+\cdots+c_n^\prime y_n^{(n-1)}=g(t).\]

This gives us $n$ equations and $n$ unknowns (where the unknowns are all functions). Solving this linear system of equations, gives us the exact value of $c_1^\prime,c_2^\prime,...,c_n^\prime $. Integrating each one, gives us the exact value for $c_i$'s. This can be done explicitly for general $L$ and $g$, as it is done in Theorem 3.7.1, and chapter 4.4. I encourage you to try to do this amongst yourself at some point for the second degree equation. It is actually a fairly straightforward computation, and doing it by hand will take away quite a bit of mystical nature of it.

These methods both require a bit of practice to get comfortable with. Problems in page 184 and 235 are great practice for method of undetermined coefficient. Problem 28-30 on page 185 is worth looking if for no other reason than the fact that we will be doing this later in the class. Problem 31,32 should give you a better feel for how these differential equations behave, and what their solutions are like. Problem 33 on page 185 and 20-22 on pages 236-237 give an alternate approach to solving these equations all together.
Similarly probelms on page 240-241 and 190 (problem 1-21) are great practice for method of variation of parameters. Problems 22-27 on page 190-1 are probably one of the most interesting sequence of problems in this book, and it is a motivation of much of modern analysis. Those of you who are interested in pure mathematics, you should definitely look at this problem. Problem 28 shows how reduction of order can be used to solve the higher order linear ODEs.

Fundamental Solutions of Linear Homogeneous Equations

2010-01-26T16:12:00.000-08:00

On Monday we mostly covered section 3.2 (generalized to higher orders). We first introduced the concept of a linear differential operator, which is a fancy name for $D[f]=a_n f^{(n)}+\cdots a_1 f^\prime + a_0 f.$ (The book uses $L$ whenever I use $D$.) Linear differential operators are special case of differential operators. What makes these linear is that $D[f+g]=D[f]+D[g]$ and $D[cf]=cD[f]$ for all functions $f$ and $g$ and all constants $c$ (this should be reminiscent of linear operators you see in linear algebra). This is easy to verify, and is left as an easy exercise.

With this notation in hand, the problems of solving linear ODE's can be stated as: given $D$ a linear differential operator and $c(t)$, find all $f$'s such that $D[f]=c$. Theorem 3.2.2 in the book says that if $y_1$ and $y_2$ are solutions to $D[f]=0$ (i.e. $D[y_1]=D[y_2]=0,$) then $c_1y_1+c_2y_2$ is also a solution to $D[f]=0$ for all constants $c_1$ and $c_2$. In the language of linear algebra, the kernel of a linear operator forms a vector space. What is slightly more difficult to prove is that the the dimension of the kernel of a linear operator of order $n$ is exactly $n$. That means if $D$ is a second order linear differential equation, then to find all the solutions to $D[f]=0$, it is enough to find two linearly independant solutions, and then their span gives us all the solutions. When $D$ is a linear operator with constant coefficients, we have a method for doing that by assuming the solutions is of the form $y=e^{rt}$, and pursuing the solution from there.

Furthermore, we showed that if we are interested in solving $D[f]=c$, then we should first solve $D[f]=0$ (i.e. if $D$ is of order $n$, find $n$ linearly independant solutions $f_1,f_2,...,f_n$ to $D[f]=0$), and then find one solution to $D[f]=c$, say $f_s$. Then the general solution is of the form $f_s+c_1f_1+\cdots+c_nf_n$ for $c_1,c_2,...,c_n$ any choice of constants.

The first half of the exercises in this section are concerned with the Wronskian, which I will talk about tomorrow. However, problems 13,14,15, 21-27 are good exercises to do. A really good exercise for you is to find second order linear differential equations that has $f(t)$ and $e^t$ as a solution, for any given $f$. Problems 28-31 cover the idea of exactness for second order equations, and 33-38 cover the adjoint equation (which is also mentioned in the wikipedia link above in a different disguise). They are definitely interested to look at, specially if you are interested in learning more about differential equations.

Higher order linear ODE,

2010-01-22T09:40:00.000-08:00

Today we looked at linear ODEs of higher order. Actually, we looked at a very specific case where the coefficients were constant (chapter 3.1 and 4.2). That is we started to look at differential equations of the form
\[ a_n{d^n\over dt^n} y+ a_{n-1} {d^{n-1} \over d_t^{n-1}}y + \cdots + a_1 {d \over dt}y+a_0 y = 0.\]
Our basic trick was to try $y=e^{rt}$ as a possible solution in this differential equation. If we do that, we get
\[ a_n r^n+a_{n-1}r^{n-1}+\cdots a_1r+a_0=0.\]
The polynomial $a_nr^n+\codts a_0$ is called the characteristic polynomial attached to the differential equation.

Bonus question: There is also a characteristic polynomial attached to a matrix . Are these two polynomials related to each other in any way?

We know that a degree $n$ polynomial usually has $n$ distinct roots. Assume that we are in that situation, and call the roots $r_1, r_2, ..., r_n$. Then we get that $e^{r_1 t}$, $e^{r_2t}$, ..., $e^{r_nt}$ are all solutions to our differential equation. One can check that any linear combination of these solutions is also a solution to our differential equation. It is a bit more difficult to prove that the general solution is
\[ y=c_1 e^{r_1t}+c_2 e^{r_2t}+\cdots c_ne^{r_n t}. \]

Problems 1-16 on page 142 are good practice to solving second order linear DEs. I do recommend problems 17, 18, 21, and 22 are very good problems that I strongly recommend you to look at. Problem 27 is a fun problem as well. Problems 11-36 of page 230 is a good practice for solving higher order linear equations. However, it is possible that some of the characteristic polynomials have complex roots in them. If that doesn't bother you, you can just solve the problems, otherwise, you can wait until we cover complex numbers. Problem 37, however, can be done by just plugging and checking.

Euler's Method,

2010-01-20T11:20:00.000-08:00

Today we talked about how a computer would approximate the solution to a differential equation $ {dy \over dt} = f(t,y)$ with initial condition $y(t_0)=y_0$, in some interval $t \in I=(t_0,t_{last})$. Throughtout this problem, we assume that there exists a $y(t)$ satisfying the differential equation and $y(t_0)=y_0$ (this is the case if, for example conditions in
The basic idea, as presented in chapter 2.7, is to divide the interval $I$ into $N$ pieces $t_0<\cdots < t_N$ Euler's method tells us to take
\[ Y_{i+1}=Y_i+f(t_i,Y_i)(t_{i+1}-t_i). \]
Intuitively, what that tells us is to move from time $t_i$ to $t_{i+1}$ with the speed of $f(t_i,Y_i)$. Algebraically, this is using the approximation of
$y'(t_i)=f(t_i,y(t_i))$ by the slope of the secant $ {Y_{i+1}-Y_i \over t_{i+1}-t_i}.$ Here is a simple code for solving the differential equation $ {dy \over dt}=ty$ with initial condition $y(0)=1$. (The code is written in SAGE, which is a derivative of Python.) As you can see, with $10$ steps, we get a descent idea on what the actual function should look like. You can probably play around with the parameters to see how well the function converges.

The analysis of how well Euler's method converges to the real answer is given in chapter 8.1 (page 447 or so), and I will go through it in some detail on Friday. The basic tool here it Taylor's remainder theorem, which states that
\[ y(t_{i+1}) = y(t_i+(t_{i+1}-t_i)) = y(t_i)+(t_{i+1}-t_i)y^\prime(t_i)+R_2(t_{i+1}), \]
where
\[ R_2(t_{i+1}) = {(t_{i+1}-t_i)^2 \over 2!} y^{\prime \prime}(s_i), \]
for some $s_i \in (t_i,t_{i+1})$. Note that $y^{\prime}(t_i)=f(t_i,y(t_i)),$ and
therefore \[y^{\prime \prime}={d \over dt} f=f_t+f_y y^{\prime}=f_t+ f_y f.\] Now, assume that $t_{i+1}-t_i=h$ for all $i$
(i.e. at each step were looking forward for small time interval of length $h$). Then we get
$\begin{aligned} |y(t_{i+1})-Y_{i+1}| &=|y(t_i)+hy^\prime(t_i)+{h^2 \over 2}y^{\prime \prime}(s_i) - Y_i - hf(t_i,Y_i) | \\ & \leq |y(t_i)-Y_i | + h |y^\prime(t_i) - f(t_i,Y_i) | + {h^2 \over 2} | y^{\prime \prime}(s_i)| \\ &=|y(t_i)-Y_i|+h|f(t_i,y(t_i)-f(t_i,Y_i)|+ \\ & {h^2 \over 2} |f_t(s_i,y(s_i))+f_y(s_i,y(s_i))f(s_i,y(s_i)|. \end{aligned}$
Assuming $f$, $f_t$, and $f_y$ are all bounded in the region we are interested in, then we can approximate the last term by some big constant $D$, independant of $h$. Similarly, we get
\[ f(t_i,y(t_i))=f(t_i,Y_i+y(t_i)-Y_i)=f(t_i,Y_i)+(y(t_i)-Y_i)f_y(t_i,W_i), \]
which means we can bound the second term with $h|y(t_i)-Y_i|D$. Therefore we get
\[ |y(t_{i+1})-Y_{i+1}| \leq |y(t_i)-Y_i|(1+hD)+{h^2D \over 2}. \]
One can now prove that $|y(t_n)-Y_n| < nh^2 M$ for all $n$, and for some $M$ (essentially $D$ times the length of the time interval) independant of the choice of $n$ (hint: this can be used using basic induction).

As for the problems in this section, I recommend you doing Euler's method by hand and calculator for few simple examples (like problem 1 and 2), just so you figure out how the algorithm works. To do more than few steps of Euler's method, I strongly recommend you using the computer (either using a programming language like python or sage, or using spreadsheet, or whichever is easiest for you.) Problems 20-23 can be done without a using a calculator. Specifically, problem 20 is a good test problem.

Exact equations and integrating factors,

2010-01-18T07:50:00.000-08:00

On Friday we covered a new method for solving a class of first order ODE's. Specifically, we called a first order ODE $M+N {dy \over dx}=0$ exact
if there exists a function $\psi$ such that $ {\partial \psi \over \partial x} = M$
and $ {\partial \psi \over \partial y} = N$. We also saw that an exact equation simplifies to $ {d\psi \over dx}=0$ (this is just chain rule in multivariables), which leads to the implicit equation $\psi(x,y)=K$ for some constant $K$. A simple calculation also shows that the above ODE is exact if and only if $ {\partial M \over \partial y} = {\partial N \over \partial x}.$ We also saw that given an exact first order ODE, we could find $\psi$ explicitly by first finding the antiderivative of $M$ with respect to $x$, which gives us $\psi$ up to a function of $y$, and then solve for $\psi_y=N$.

Today, we followed this up with introducing the integrating factor. Specifically, if $M+Ny'=0$ is not exact, we asked if there is a function $\mu$ such that it will make $\mu M+\mu N y'=0$. Checking if this equation is exact leads to the first order PDE \[ M \mu_y - N \mu_x +(M_y-N_x)\mu = 0.\]
Unfortunately, this PDE is pretty difficult to solve in general, so for us to actually find such a $\mu$ we usually need to make a simplifying assumption. Specifically, we usually assume that $\mu$ is a function of one variable so we will get an ODE rather than a PDE. Examples of such assumptions are assuming $\mu(x,y)=f(x)$ (what we covered in class, and is covered in the book), $\mu(x,y)=f(y)$ (problem 23), or even $\mu(x,y)=f(xy)$ (problem 24).

Despite all of the symbols running around in this section, the actual method of solving exact equations and finding integrating factors is not that bad. I strongly encourage you to try the problems in this section, to get the handle of this subject. Problems 1-16 are good exercise for solving exact equations. Provlems 17 and 18 are a bit more theoretical, and will make good proof questions. Problems 18-22 should be treated as a warm up for the integrating factor, while 25-31 are more serious. I strongly encourage you to solve problems 23 and 24.

A fair (but challenging) problem for the midterm/exam is the following:

Solve $x^2 y^3+x(1+y^2)y'=0$. (Hint: This ODE has an integrating factor of the form $\mu(x,y)=f(xy^3)$.)

Existence uniqueness and autonomous equations.

2010-01-13T08:09:00.000-08:00

Today we covered sections 2.4 and 2.5 of Boyce and DiPrima. The big result in 2.4 are theorems 2.4.1 and 2.4.2, which gives sufficient condition for a first order ODE to have a unique solution. As I commented in class, the proof of 2.4.1 is much more straightforward, and I encourage you to try proving the result by yourself, and figuring out how the conditions in the theorem are chosen. The proof of theorem 2.4.2 is postponed until later section, however, you should try to understand what the theorem is stating. Examples following the theorem are a good read to try to get familiarity with this theorem. We did not talk about Interval of Definition explicitly, however reading that part (is a fairly quick read) is not a bad idea. We'll deal with interval of definition for linear equations later in the course (when dealing with higher order linear ODEs specifically.)

Section 2.5 deals with autonomous differential equations. These are equations of the form $ {dy \over dt}=f(y)$. Another way of stating this is that the derivative of $y$ depends only on $y$, and aside from that it is independant of $t$ (of course, $y$ is a function of time itself, so the derivative changes over time, however that's the only dependance.) The book focuses their examples on population dynamics, and as result they define few terms that are related that subject. For example, rate of growth, intrinsic growth rate, environmental carrying capacity, etc., they all have their roots in population dynamics.

The goal of this section for us is to be able to make a qualitative plot of the function $y(t)$ without too much work (we assume $f$ is continuous). To that end, the useful definitions for us are equilibrium solution, asymptotically stable solution, and unstable equilibrium solution. Our approach to plotting solutions of $ {dy \over dt}=f(y)$ is to first find the equilibrium solutions of this differential equations. These are the ones where $f(y)=0$. We can plot those very easily ($y=K$ where $K$ is a root of $f$). This divides our graph in few regions ($r+1$ where $r$ is the number of roots of $f$) and each region we decide if $y$ is increasing or decreasing (this is done by checking if $f(y)>0$ or $f(y)<0$ in this region). And based on that we can draw the plot of $y(t)$, at least qualitatively. Note that there are two types of equilibrium solutions at this point: the ones that attract other solutions, and the ones that repel other solutions, as $t \rightarrow \infty$. We call the attracting ones stable, and the repelling ones unstable.

A word of warning, all the plots of y versus t in this chapter are terrible! The first person who points out why that's the case, gets a bonus point.

As for problems in the book, in section 2.4, problems 1-12 gives you a good practice for how theorems in the book apply to differential equations. Problem 22 gives a nice example of how things go wrong with the theorems don't apply. Problem 24-26 gives a much cleaner way of looking at solutions to linear equations (this will be very useful in later chapters). Problems 27-31 cover an interesting non-linear equation that can be reduced to linear equation, and 32 and 33 looks at what happens when we have a linear ODE with discontinuous coefficients. In section 2.5, 1-13 are good practice for plotting autonomous equations. problem 7 shows that it is possible for an equilibrium solution to be stable from one direction and unstable from the other one. Problem 14 gives an algebraic method for deciding stability, and problems 15 onward are different models that lead to autonomous DE.

Modeling,

2010-01-11T06:58:00.000-08:00

Today's lecture, we saw examples of problems that can be modeled using differential equations. The examples I covered were mostly taken from examples 1 and 2 of section 2.3. At this point, we've seen three very different problems that were modeled using differential equations: the two from today's lecture, and the Newton's Law of cooling from first lecture. Note that for Newton's law of cooling, we gave the differential equation and we started from there. If we were going to write down this law in english, it would probably say something like "the rate of change of temperature is proportional to the difference between the temperatures". In the exercises in the book, sometimes the differential equation is given. In those cases, try to convert the differential equation to a reasonable english sentence (not always possible).

Example 4 of section 2.3 is the example I breifly mentioned toward the end of the class, and you should read it. The best part of example 4 is that the differntial equation that you end up with is more complicated than the previous examples. Specifically, we get \[v {dv\over dx}=-{g R^2 \over (R+x)^2}.\] This is both nonlinear, and both variables show up on the other side (i.e. it is not autonomous). Example 3 is another somewhat reasonable example of a real life problem that leads to more complicated differential equations.

This section is filled with word problems, that should give you great practice for modeling real life problems using differential equations. I strongly encourage you to read over some of the later questions in this chapter for practice. Problems such as 13, 25, 29. and 32 should be pretty challenging.

Next class we'll cover sections 2.4 and 2.5.

Linear equations and Integrating factors

2010-01-08T09:49:00.000-08:00

Today we looked at the method of integrating factors for solving first order linear ODE's (section 2.1). The basic idea is that if you have equation $ {dy \over dt}+py=g$, you can find a function $\mu$ such that $\mu {dy \over dt}+\mu p y={d \over dt}(\mu y)$. Using product rule for differentiation we get that that $ {d \over dt}(\mu y)={d \mu\over dt} y+{dy \over dt} \mu$, and
therefore such an $\mu$ satisfies $ {d\mu \over dt}=p\mu$ which is a separable differential equation. Any solution to this differential equation is called an integrating factor. After finding an integrating factor, we can simplify our differential equation to \[ {d \over dt} (\mu y) = \mu g, \]
which can be solved using basic integration.

It is important to note that this is not the only method for solving linear first order ODE's. Another option (known as variation of parameters), first assumes $g=0$, and solved $ {dy\over dt}+py=0$. We can solve this using separation of variable, and the general solution will be $y=CY(t)$, for some contant $C$. Now, when solving $ {dy\over dt}+py=g$, one assumes the solution is of the form $C(t)Y(t)$, where $C$ is now a function of time. Plugging this equation back in the differential equation, we get a simple integral that gives us what $C$ must be. (Exercise 38).

It is also important to note that the method of integrating factors can be used to solve few other problems that are not linear as well. For example, I strongly encourage you all to solve the differential equation $$ {d \over dt}y+2y=ty^3 $$.

Most of the problems in this sections are straightforward application of the method in question. (Problem 26 is an example of how knowing the method can help you see a quick shortcut.) I also really like problems 27-37, since they give you some insight on how solutions to linear ODEs behave.

Finally, there were few questions after class about other ideas for solving linear ODEs. I should emphasize that there are many methods for solving linear ODEs in general, and if you have a method in mind, then you should see if you can make it work in general. My favourite method (guess and check, so it won't work all the time) is to make an educated guess for what the solution is going to look like, and the substitude that form of the function in the equation and see if you can solve for the unknown parameters in question. The more experience you have in solving ODEs, the more likely it is for this method to work.

Definitions and Separable Equations,

2010-01-06T07:17:00.000-08:00

Today we covered sections 1.3 and 2.2. In the class I defined an ODE to be an equation of the form \PHI(f,f',f'',...,f^(n),t)=0. I probably mentioned this at some point, but it doesn't hurt to emphasize that f is a funtion of t, in such equations. Unfortunately, dues to laziness, it is easier to drop the t dependance from all of the f's above (you should try writing the equation with f(t) replacing f everywhere few times though). Also, one concept that I did not mention from the book is the concept of linearization (which is approximating your DE with a linear DE). It is a very powerful tool, and I encourage you to read that part of the book (pages 20-21 in my text). Section 1.3 ends with a discussion about existence and uniqueness of the solution, unfortunately we will not discuss this subject much in this class, but if you want to read more about that the wikipedia pages on ODE and PDEs are good starting point for what is known (they are good pages to read anyways). The problems in section 1.3 should give you lots of practice in figuring out the type of a differential equation you are dealing with.

Section 1.4 is definitely a fun read, although you might want to take some of the anecdotes with a grain of salt (they do make great stories though :D).

We will get back to section 2.1 on Friday, but we looked at section 2.2 next. (I'm very confused on why the book does 2.1 before 2.2, to be honest). The book proves the separation of variable technique more carefully than we did in class , and if those dangling differentials (df's or dt's who were all alone) bothered you, you should read the book's treatment which deals with this issue a bit better. The pictures of this section are also very pretty, so you can enjoy them. Also, we skipped the range of solution. We will touch back on that on Friday. As for problems in this section: problems 1-29 are good practices for this technique (specially if you can plot the solutions). Problem 30 and later show how some equations that don't look separable can be solved using the same technique.

Math 2C03 Blog,

2010-01-06T06:43:00.000-08:00

This is the first post for Math 2C03. The plan is to make a blog post after almost every class with information that might be useful for that lecture. The first post will be about the first lecture, which was mostly about administrative stuff. I still haven't recieved any emails that students can't attend my office hour, so it either means that nobody has any conflict with the two time slots (unlikely in this sized class), or that the ones that do have a conflict haven't got around sending me an email about it. I will probably mention this again during Friday's lecture.

Math wise, we covered section 1.2 of Boyce and DiPrima (called the book in most posts), and started section 1.3. I strongly encourage the students to read section 1.1 as well (it is reasonably easy to read, and does a good job for motivating the problems we will deal with in Math2C03). The problems in section 1.2 are definitely worth looking at. Here is a list of possible questions: 3, 5, 8, 11, 15, 16, 19.