Differential Equations Course Notes

General Course Introduction and Expectations

The instructor expresses openness to providing online resources if students struggle but encourages students to find their own resources as well.
Creating additional problems beyond those covered in class is highly beneficial for learning.
The instructor emphasizes that tests will cover exactly what is taught in class.
Calculus 2 is important because it builds skills needed to solve differential equations, particularly related to entropy.
Calculus 1 introduces anti-derivatives and geometry, which are foundational concepts.
Polygons are shapes created by lines without curves and when curves are involved, problem-solving becomes more complex.
Antiderivatives are another key concept that will be used.
There are two types of solutions: general solutions and particular solutions.
- General solutions contain a variable c that can change with the scenario.
- Particular solutions provide one and only one specific solution.
Linear algebra concepts such as unique solutions and linearly dependent/independent philosophies are relevant to the course.
The goal of the course is to find either general or particular solutions to differential equations.
- Example: c = 1 when substituted into a general solution, leads to a particular solution.
The course will build popular skills for solving differential equations effectively.
Knowing which approach to use for identifying the original function is crucial.

Calculus Review and Differential Equations

Echoing previous knowledge of differentials and differential equations from calculus one is important.
Having two initial values is required to determine a panel of what will be done in the class.
Practice is essential for mastering the material.
Understanding why someone might miss a part or overestimate/underestimate is important. The Simpson's rule is mentioned.
Example: Given \frac{3}{2}x^2 + 1, which requires further steps to find the function itself, illustrating the need for another constant, d, because constants c can vary (e.g., between parabolas and lines).
To find a particular solution of a cubic function, the value of d must be known, which can be found using initial values.
Example: y = \frac{1}{2}x^3 + x + 3 is presented as a particular solution derived from a second-order function.
Some methods are more effective than others, and functions being continuous at the boundary [a, b] is a requirement for antiderivatives noting that partial continuity is acceptable because it can be broken down.
The ability to reach conclusions quickly is valued, and skills are built to think several moves ahead.
All math notes will be required for every test.
If an equation lacks an x, one can take calculus two or graph the function in the interval [a, b] using methods like Simpson's rule or Riemann sums, which can be tedious. Riemann is credited with creating the Riemann sum.
Remembering McLaurin series is necessary, as it provides a polynomial sequence which you can easily find the antiderivative over.

Integral Factors and Differential Equations

Thinking about u-substitution is important, here and there, and integral factors serve the same purpose.
Integral factors were proven and produce scenarios like x^2 or x.
Newton's Law of Cooling states that the rate of change of temperature is proportional to the difference between the object's temperature and its surroundings. This provides an example of differential equations encountered before this course.
Textbooks on differential equations typically cover first-order, second-order, and multi-order differential equations, along with techniques to find the original functions.
Newton's Law of Cooling is expressed as the time rate of change, which relates to differentiation indirectly learned in calculus one when rate of change was introduced.

Solving Differential Equations

At the beginning, we will look at what a possible solution to a differential equation may look like. Also what relation to a given differential equation is a valid solution.
Example demonstrating function t = cu, first derivative is \frac{t}{cu}. If the first derivative can be rewritten with respect to the original function (pre-calculus identity), a first-order differential equation can be found.
The function and its derivative are related allowing the whole chunk of y to be kept in it. Rewriting the derivative:
- Original rewrite: y = e^{-9x}
- The derivative : y' = -9e^{-9x} = -9y
While finding a solution through trial and error might work, it is not practical.
Find the derivatives and see if the function still exists in it as another method.
If a function is known, one can find possible derivatives to see if the function still exists within it, indicating that the function is a solution to that differential equation.
- Give an example solution for the differential equation, with a solution x. The derivative of x is 1. The general solution would be x + c.
Why is the derivative of e^x also e^x? It can be verified using the definition of a limit with e.
One example of students performing 50 derivatives only for one to get it wrong.
If angles are the same, with sine and cosine you go twice. If you want the same collisions with the sine itself, it has to go four times.
As many times you go, you'll have to have the nth derivative of that order.
Given that the general solution is a dependent general solution, there are infinite number of a similar kind.
The first chapter includes identifying possible solutions of a given differential equation.

Key Concepts and Course Policies

Distinction between general and particular solutions is emphasized; particular solutions require initial values.
Checking if y1 + y2 is a solution, you wanna check at home y1 - y2. Additionally any multiples of the equation.
The method to find the solution to the derivatives is going to the derivative to the given function, then substitute into the equation, and see if it matches.
A strong understanding of derivatives and antiderivatives is essential for this course.
Implicit differentiation is introduced as a technique, using cross multiplication to simplify equations.
The appropriate derivative, combined with an initial value, helps solve the differential equal.
Right away, you still see solutions in your domain which I will always take as an answer.
Nice general fact involved in the problem is trial and error method, our outcome is a polynomial.
Our course will will train us ways to learn them more complex for differential solutions, such that it yields general and particular equations. So don't work on guess, it is just to tedious.
You don't have to go find the integral factor when you find the solution, you have to just find y2 if it's a solution. Then go home and check y1 - y_2, or if the answers exist with a scalar.
Practice similar questions on page 8 and 9 in the textbook.
Homework is due every Monday, with questions assigned on Tuesdays and Thursdays, to be submitted in person.
Assignments are taken seriously, and missing a test, even with perfect scores on other assignments, can prevent earning a top grade.
Test dates are tentative but should be adhered to if possible.
Reach out by Monday, the day the homework's due date, if you need any questions answered.
Homework sent electronically is acceptable in extraordinary circumstances (e.g., train delays).
No AI is allowed on the homework, it should be all written.
Homework serves as test review problems.
All test material comes from class discussions and homework assignments.