Differential equations

from University of Tokyo 1S1 Mathematical Science Foundation: Differential and Integral Calculus

Differential Equations

• Equations in high school: Algebraic equations
  • Like $x^2+x+1=0$
• Differential equations
  • Like $y^{\prime \prime }+y^{\prime }+y=0$
  • “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”
• 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?
  • $y^{\prime }=f(x)g(y)$ 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)
• 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?

    • 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?
  • 3.2 Solution methods for first-order homogeneous equations
    • Self-study (blu3mo)