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# Math 15 Ordinary Differential Equations
This page is more for myself to stage and plan. But you might find it useful as well. You should always attend lecture and take notes.
Summer 2025
JUNE 23, 2025 to JULY 31, 2025
Syllabus -- TBD
Class Schedule
08:00AM-10:05AM .MTWR..
MSB 203
Weeks 1 - 6
Main Topics
1. Classification and basic terminologies of ODEs, examples.
2. General theory of first order ODEs, basic strategies of finding solutions to first order ODEs.
3. General theory of linear ODEs
4. Basic numerical method (Euler's method), computer implementations.
5. Power series method
6. Laplace transform
7. Applications.
Exams
Exam 1 Week 2 Thursday 7/3 In-class
Exam 2 Week 4 Thursday 7/17 In-class
Final exam Week 6 Thursday 7/31 In-class
See Canvas for class notes taken in class.
Tentative Daily plan
**Week 1**
[[1-teaching/smc-summer-2025-math15/w1.1|6/23 Monday]]
Sponsored by the letter S, and the number $\frac{1+\sqrt{5}}{2}$.
- Modeling to get a DE / Snow fall !
- Using FTC.
- Classification of ODEs.
- Solutions and interval of definition.
[[1-teaching/smc-summer-2025-math15/w1.2|6/24 Tuesday]]
Sponsored by the letter M, and the number $\ln(\frac{68}{20})$.
- Discrete simulation of snow fall.
- Separation of variables.
- Method of separation and its proof.
- Differential form expression.
- Implicit vs explicit solutions
- 1-parameter family of solutions
- Abuse of constant
- Newton law of cooling and heating / M is for .... Murder !
- First-order monic linear ODEs.
[[1-teaching/smc-summer-2025-math15/w1.3|6/25 Wednesday]]
Sponsored by the letter P, and the number $\frac{1}{e^{3}}$.
- Constant solutions.
- Initial value problem (IVP)
- Do solutions even exists? If so, are they unique? M is for ... Monsters !
- Existence and uniqueness theorem
- Peano's existence theorem
- Picard's existence and uniqueness theorem
- Picard iterates
[[1-teaching/smc-summer-2025-math15/w1.4|6/26 Thursday]]
Sponsored by the letter E, and the number 3.
- 1-parameter family of curves
- Slope fields.
- Differential forms.
- Exact equations and exactness criterion.
- Non-exact case: Integrating factor.
**Week 2**
[[1-teaching/smc-summer-2025-math15/w2.1|6/30 Monday]]
- Substitution methods
- Higher order substitution
- Homogeneous of same degree
- Linear substitution
- Bernoulli
- Orthogonal family of curves
[[1-teaching/smc-summer-2025-math15/w2.2|7/1 Tuesday]]
- What the curve !? Parabolic reflectors.
- Autonomous system and phase line analysis of equilibrium solutions.
- Logistic equation.
- Salt water example
[[1-teaching/smc-summer-2025-math15/w2.3|7/2 Wednesday]]
- Numerical methods
- Euler's method
- Computer implementation with JavaScript.
- Euler's formula
7/3 Thursday **EXAM 1 IN-CLASS**
**Week 3**
[[1-teaching/smc-summer-2025-math15/w3.1|7/7 Monday]]
- Constant coefficient homogeneous linear ODEs.
7/8 Tuesday
7/9 Wednesday
7/10 Thursday
**Week 4**
7/14 Monday
7/15 Tuesday
7/16 Wednesday
- Generating functions.
7/17 Thursday **EXAM 2 IN-CLASS**
**Week 5**
7/21 Monday
- Laplace transform -- its definition.
- Laplace transform of basic functions.
- Linearity.
- Derivative rules.
- Solving differential equations with Laplace transform.
7/22 Tuesday
- Sufficiency conditions for Laplace transform to exist.
- Gamma function.
- System of linear DE.
7/23 Wednesday
- Unit step function.
- Exponential shift rules.
- Integral rules.
- Delta function introduction.
7/24 Thursday
- Delta function and unit impulse.
- Convolution and convolution theorem.
- Analogies between Laplace transform and generating functions.
**Week 6**
7/28 Monday
- Proof of the convolution theorem.
- Properties of convolution.
- Examples.
7/29 Tuesday
- Examples.
7/30 Wednesday
- Class party / photo.
7/31 Thursday **FINAL EXAM IN-CLASS**
- If time permits: More numerical methods
- Modified Euler's method (2nd order method)
- Runge-Katta( 4th order method)