from University of Tokyo 1S1 Mathematical Science Foundation: Differential and Integral Calculus
Differential Equations
- Equations in high school: Algebraic equations
- Like
- Differential equations
- Like
- “Equations of motion” are also differential equations
- They are also discussed in mechanics (blu3mo)
- Unlike algebraic equations, the solutions to differential equations are functions
- If we can define the algebra of functions, we can define something like algebraic equations of functions (blu3mo)
- If we consider function composition as multiplication, it would be like $f(f(x))+f(x)+1=0”
- If we can define the algebra of functions, we can define something like algebraic equations of functions (blu3mo)
- Terminology is self-study (blu3mo)
- Separable type
- First-order ordinary differential equations
- What is ordinary differential calculus?
- Is it the differentiation with one variable up to high school?
- What is ordinary differential calculus?
- is called separable type
- We learned this in high school (blu3mo)
- Is it the one with easy differentiation calculations? (blu3mo)
- Just move y to the left side and x to the right side, then integrate
- Note: When 1/y appears, we need to consider the case of y=0 as well
- If it’s not separable type, we use the Integrating Factor Method
- I don’t know this term (takker)
- We learned this in high school (blu3mo)
- First-order ordinary differential equations
- First-order homogeneous linear differential equations
- What is an nth-order linear differential equation?
- An equation where only y, its nth derivative, and constants appear?
- I don’t quite understand why it is called linear (blu3mo)
- TODO: What has linearity?
- What is a homogeneous linear differential equation?
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A linear differential equation in which all terms contain the unknown function or are equal to 0
- Is it called a homogeneous differential equation when there are no terms with x alone?
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- 3.2 Solution methods for first-order homogeneous equations
- Self-study (blu3mo)
- What is an nth-order linear differential equation?