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.
Wednesday, March 31, 2010
Saturday, March 13, 2010
Next assignment
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.
Wednesday, February 24, 2010
Laplace transform
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:
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.
- $ \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}.$
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.
Friday, February 19, 2010
midterm on Tuesday,
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
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
Tuesday, February 9, 2010
Non homogeneous equations
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.
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.
Tuesday, January 26, 2010
Fundamental Solutions of Linear Homogeneous Equations
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.
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.
Friday, January 22, 2010
Higher order linear ODE,
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.
\[ y=c_1 e^{r_1t}+c_2 e^{r_2t}+\cdots c_ne^{r_n t}. \]
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.
\[ a_n{d^n\over dt^n} y+ a_{n-1} {d^{n-1} \over d_t^{n-1}}y + \cdots + a_1 {d \over dt}y+a_0 y = 0.\]
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.
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