Lecture on First-Order Ordinary Differential Equations

Jul 30, 2024

Lecture Notes: Differential Equations

Introduction

  • Assumptions:
    • Attended recitation
    • Know how to separate variables
    • Construct and solve simple models with differential equations
    • Reminder: Book has a good explanation for basics

First-Order Ordinary Differential Equations (ODEs)

  • ODE: Ordinary Differential Equations
  • Form: Isolate the derivative of y with respect to x on the left side, write everything else on the right side
  • Example equations:
    • Solvable by separation: ( y' = \frac{x}{y} )
    • Not separable but solvable: ( y' = x - y^2 )
    • Not solvable in elementary functions: ( y' = y - x^2 )

Analytical vs. Geometric Views

  • Analytic Method: Write equation as ( y' = f(x, y) ) and solve for functions using elementary methods
  • Geometric Method: Use direction fields and integral curves
    • Direction Field: Plane filled with small line elements with slopes defined by the ODE ( f(x, y) )
    • Integral Curve: Curve tangent to line elements at each point, representing solutions of the ODE

Fundamental Geometric Concepts

  • Integral Curve Theorem: ( y_1(x) ) is a solution of ( y' = f(x, y) ) if and only if its graph is an integral curve of the associated direction field
  • Drawing Direction Fields:
    • Computer Method: Equally spaced points calculated with ( f(x, y) )
    • Human Method: Plot isoclines for specific slopes first, then draw line elements

Examples and Applications

  1. Example 1: ( y' = -\frac{x}{y} )
  • Isoclines: ( y = -\frac{x}{C} )
    • E.g., C=1: ( y = -x )
    • Line elements have slope C
  • Integral Curves: Circles centered at origin
  1. Example 2: ( y' = 1 + x - y )
  • Isoclines: ( y = x + 1 - C )
  • Geometric Approach: Draw parallel line isoclines and fill in line elements based on slope
  • Integral Curves behave asymptotically

Important Principles

  • Two integral curves:
    • Can't Cross: Only one defined slope at each point
    • Can't Touch: Uniqueness theorem ensures no tangent intersection
  • Existence and Uniqueness Theorem:
    • One and only one solution through any point (x0, y0) if certain conditions are met
    • Conditions:
      • ( f(x, y) ) continuous near the point
      • Partial derivative ( \frac{\partial f}{\partial y} ) also continuous

Final Example: ( y' = \frac{1 - y}{x} )

  • Separation of Variables: Solutions are lines with intercept 1 and variable slopes
  • Existence and uniqueness can fail if conditions (continuity of ( f(x,y) )) are not met along certain lines (e.g., x=0)
  • Reminders:
    • Not all points guarantee existence and uniqueness of a solution, especially if ( f(x, y) ) is undefined at those points

Conclusion

  • Emphasized the importance of geometric methods to understand differential equations
  • Encouraged practice through computer exercises and manual drawing of direction fields and isoclines