Lecture Notes: Substitution Techniques in Differential Equations

Introduction

• Techniques for solving differential equations (DEs) that don't fit basic models like:
  • Basic integration
  • Separable equations
  • Linear first-order differential equations
• Focus on substitution techniques to convert non-recognizable equations into solvable forms.
• Five categories of substitution techniques:
  1. Homogeneous equations
  2. Obvious substitutions
  3. Bernoulli equations
  4. Embedded derivatives
  5. Reducible second-order DEs
• Additional category: Exact equations (to be discussed later)

Homogeneous Equations

• Substitution method to convert into known techniques (basic integral, separable, or linear equations).
• Homogeneous equation: Function where each term can be expressed as a fraction (y/x).
• Substitution Plan:
  • Let V = y/x
  • Solve for y in terms of V and x
  • Differentiate implicitly: Use product rule and chain rule
  • Substitute into the differential equation
  • Solve using known techniques, then back-substitute to find y.

Steps:

1. Structure DE as dy/dx = function of (y/x)
2. Make substitution V = y/x
3. Solve for y: y = Vx
4. Differentiate: dy/dx = x(dV/dx) + V
5. Substitute and simplify to form a solvable DE
6. Solve using known techniques (e.g., separable equations)
7. Back-substitute to express solution in terms of y

Example Problems

• Example 1: Given a DE, solve using homogeneous substitution.

  • Solve initial DE for dy/dx
  • Identify instances of y as y/x
  • Substitute V = y/x and differentiate
  • Solve resulting separable differential equation
  • Substitute back to express final result in terms of y

• Example 2 & 3: Process repeated to reinforce technique with different equations.

Additional Considerations

• Domain Restrictions:
  • Must account for domain limitations when dividing by terms involving x or y.
  • Domain restrictions arise naturally from the requirement to avoid division by zero and handle square roots correctly.
• Homogeneous equations often yield separable equations.
• Techniques may vary, and sometimes multiple methods can apply to solve a DE.

Conclusion

• Homogeneous equations offer a structured method to solve DEs fitting the (y/x) pattern.
• Important to structure correctly to ensure all instances of y are represented as (y/x).
• Substitution techniques allow the transformation of complex DEs into forms solvable by known methods, enhancing problem-solving flexibility.

Next Steps

• Practice additional examples to solidify understanding.
• Explore other substitution categories in upcoming lectures/videos.