- Here is what I wrote during the final exam. Please let me know if you find errors in my solutions.
- Here is a comprehensive list of topics for the final exam.
- This is what I wrote during the exam.
- I gave a review of the power of the Laplace transform today. Details are covered in Sections 5.1, 5.2, 5.3 and 5.4.
- The assigned exercises for Section 5.1 are 1, 3, 4, 7, 9, 10, 12, 13, 15-24, 25, 27.
- The assigned exercises for Section 5.2 are 1-6, 9, 11, 13, 18, 21, 23, 26, 27.
- There are 36 exercises in Section 5.3. More you do, better you will master finding the inverse Laplace transform. Completing the square and partial fractions are the main tools.
- In Section 5.4 you should do a selection of exercises from 1-36. Do 35 and 36. These are interesting exercises.
- We will start Chapter 5 on Friday.
- The assigned exercises for Section 5.1 are 1, 3, 4, 7, 9, 10, 12, 13, 15-24, 25, 27.
- The assigned exercises for Section 5.2 are 1-6, 9, 11, 13, 18, 21, 23, 26, 27.
- We started Section 4.7 Forced harmonic motion today. The assigned problems for this section are 1-7, 9-11, 12-15, 16-19, 22, 23, 24, 25, 26, 27, 31-38, 43, 44, 45.
- Here are some solutions of problems in Section 4.7 (updated Nov. 15). I might add few more problems.
- Here are few solved problems from Section 4.4.
- We will do Section 4.5 Inhomogeneous equations; the method of undetermined coefficients tomorrow. The assigned problems for this section are 1-4, 10-13, 18-21, 24-29.
- This is what I wrote during the exam.
- We will do 4.4 Harmonic motion tomorrow. All problems in this section are assigned.
- The second exam is on Tuesday. It will cover Sections 2.9, 3.3, 3.4, 4.1, 4.3, and the handout on complex numbers.
- Here is a link to the Math center web page that lists all math fellows who have taken a course in differential equations. Those math fellows should be able to help you out. Please, also, take advantage of my office hour, 10am on MTThF, or make an appointment.
- Important concepts from Section 4.1.
- Understand Example 1.9 on page 166.
- The vibrating string example: Understand the derivation of the equation (1.13) on page 167.
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The concept of a second order linear differential equation:
\[ y^{\prime\prime}(t) + p(t) y^{\prime}(t) + q(t) y(t) = g(t) \]
The functions $p(t), q(t)$ and $g(t)$ are called coefficients. The coefficient $g(t)$ is called the forcing term. If $g(t)=0$, then the equation is said to be homogeneous:
\[ y^{\prime\prime}(t) + p(t) y^{\prime}(t) + q(t) y(t) = 0 \]
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The Wronskian of functions $f(t)$ and $g(t)$ is the following function:
$ W(f,g)(t) = f(t)g^{\prime}(t) - f^{\prime}(t) g(t) $
If the Wronskian is non-zero at some point in an interval, then the functions $f(t)$ and $g(t)$ are linearly independent on the interval.
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Two functions $y_1(t)$ and $y_2(t)$ form a fundamental set of solutions of a homogeneous equation
\[ y^{\prime\prime}(t) + p(t) y^{\prime}(t) + q(t) y(t) = 0 \]
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$y_1(t)$ and $y_2(t)$ are solutions of the homogeneous equation $y^{\prime\prime} + py^{\prime} + qy = 0$.
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The Wronskian $W(y_1,y_2)(t) = y_1(t)y_2^{\prime}(t) - y_1^{\prime}(t) y_2(t)$ is nonzero.
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When we are solving a homogeneous equation
\[ y^{\prime\prime}(t) + p(t) y^{\prime}(t) + q(t) y(t) = 0, \]
we first find a fundamental set of solutions $y_1(t)$ and $y_2(t)$. Then all the solutions are given as linear combinations of $y_1(t)$ and $y_2(t)$, that is as,
$ C_1 y_1(t) + C_2 y_2(t). $
This formula is the general solution of the homogeneous equation. Here $C_1$ and $C_2$ are arbitrary constants.
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When we are solving an initial value problem
\[ y^{\prime\prime}(t) + p(t) y^{\prime}(t) + q(t) y(t) = 0, \ \ \ \ y(0) = y_0, \ \ \ y^{\prime}(0) = v_0, \]
we first find the general solution, then we calculate $C_1$ and $C_2$ from the equations
$C_1 y_1(0) + C_2 y_2(0) = y_0$ and $C_1 y_1^{\prime}(0) + C_2 y_2^{\prime}(0) = v_0.$
This gives us the particular solution which satisfies the initial conditions.
- The assigned problems for Section 4.1 are 1-8 and 10 - 13.
- The assigned problems for Section 4.3 are 1 - 29.
- Today we started Section 4.1. The assigned problems are 1-8 and 10 - 13. On Thursday we will do Section 4. 3. The assigned problems are 1 - 29.
- Today we talked about complex numbers. Here is a short summary with some exercises. Do the exercises for homework. One application of the polar form of complex numbers is in solving circuit equations which involve oscillating voltages in Section 3.4.
- Today we started Section 3.4. All problems are assigned. Some solutions are here.
- Today we did Section 3.3. The assigned problems are 1 - 10. The solutions are here.
- An extended version of Exam 1 is now a homework due in class on Monday, October 19.
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Today we touched Section 2.9: Autonomous equations and stability. The assigned problems for this section are 7, 8, 9, 10, 11, 13, 14, 17, 18, 19, 21, 23, 24. Example 2.9.3 is somewhat complicated. Ignore it and do the tank example that I did in class.
Remember, $V$ is the volume of the tank, $a$ gal/min of brine enters the tank and the brine leaves the tank with the same rate, $c$ is the concentration of salt in the incoming brine in lb/gal. Denote by $S(t)$ the amount in lb of salt in the tank at time $t$. The differential equation governing this process is
\[ S'(t) = c\, a - \frac{a}{V} \, S(t). \]
The common sense tells us that if there are $S(0) = c\,V$ pounds of salt in the tank at time $t = 0,$ then there will be no change in the amount of salt in the tank. That is $S(t) = c\,V$ is an equilibrium solution. Again, common sense tells us that if $S(0) > c\,V$, then $S(t)$ will decrease and in the limit it will get closer and closer to $c\,V$. Similarly, if $S(0) < c\,V$, then $S(t)$ will increase and in the limit it will get closer and closer to $c\,V.$ It might be informative to rewrite the differential equation as
\[ S' = \frac{a}{V} \bigl( c\, V - S \bigr). \]
From this equation we can see that $S'$ is positive, that is the function $S$ is increasing, whenever $S < c\,V$ (in this case $c\,V - S > 0$). Similarly, $S'$ is negative, that is the function $S$ is decreasing, if $S > c\,V$ (in this case $c\,V - S < 0$).
- I am using a third party website to display the formulas above. My experience it that this works best in Mozilla Firefox browser. In Chrome and IE few times I had problems displaying formulas. Usually formulas display fine next time I visit, but refreshing the page did not help. Please let me know about your experience.
- Today we did Section 2.5: Mixing Problems. The assigned problems for this section are 1,2, 4 - 7, 9, 12 with solutions. You can also use Mathematica to verify these solutions. Please let me know if some of the stated solutions are wrong.
- Three solutions in Section 2.3 were wrong. The corrected solutions are here.
- Today we did Section 2.4: Linear Equations. The assigned problems for this section are 1-10, 13-19, 21 with solutions. You can also use Mathematica to verify these solutions. Please let me know if some of the stated solutions are wrong.
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I corrected the solution of Problem 4 in Section 2.3.
Here is a simple
Mathematica file which I used in class. - The animations below are inspired by Section 2.3. In Section 2.3 we learned the formula for the position of a ball thrown vertically into the air.
Place the cursor over the image to start the animation.
- In the first animation below I added a vanishing trace to better convey a sense of motion.
- To make even more interesting animation I consider a baton thrown vertically into the air and which rotates about its center. For the center of the baton I use the equation of the position of the ball. Then, the endpoints of the rotating baton are determined using the parametric equation of a circle learned in Math 124. Again, I added a vanishing trace of the rotating baton to better convey a sense of motion. The trace consists of several (or rather many) previous positions of the baton which fade as the baton is getting further away.
Place the cursor over the image to start the animation.
- Solutions of the assigned problems from Section 2.3. Please let me know if you find that some of these solutions are wrong.
- Today we will do Section 2.3: Models of Motion. We will cover the first part only, concluding with Example 3.8. Here is the list of homework exercises for this section: 1-6, 8, 10, 12.
- Solutions of the assigned problems from Section 2.2. You can also use Mathematica to verify these solutions.
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Mathematica is a powerful computational software. You can findMathematica among Math applications on all computers on campus. -
Mathematica files are called notebooks, extension nb. To use the notebook in the item below, right-click on the underlined word "Here"; in the pop-up menu that appears, your browser will offer you to save the file in your directory. After saving the file you can open it withMathematica . -
Here is a
Mathematica file with several examples of direction fields. Read the file from the beginning and follow the instructions. If you have questions please ask. -
Here is a notebook with several examples of how to solve differential equations using
Mathematica .
- Today we will start Section 2.2: Solutions to separable equations. Here is the list of homework exercises for this section: 1-5, 8, 9, 13-17, 19, 20, 22, 24-27, 34, 35, 36, 37, 40