Elimination of Arbitrary Constants in Differential Equations

Introduction

• Topic: Elimination of arbitrary constants in differential equations.
• Importance: Necessary for obtaining the differential equation from a given equation.
• Relevance: Useful for studying families of curves in the next lesson.

Goal

• To eliminate arbitrary constants from given equations to derive the corresponding differential equations (DE).

Examples

Example 1:

• Equation:
  [ y = ax^2 + bx + c ]
• Number of Constants: 3 (a, b, c)
• Derivatives:
  • First Derivative: [ y' = 2ax + b ]
  • Second Derivative: [ y'' = 2a ]
  • Third Derivative: [ y''' = 0 ]
• Differential Equation: [ y''' = 0 ]

Example 2:

• Equation:
  [ x^3 - 3x^2y = c ]
• Number of Constants: 1 (c)
• Derivatives:
  • First Derivative: [ 3x^2 - 3x^2y' - 6xy = 0 ]
• Simplified DE: [ xy' - 2y = 0 ]

Example 3:

• Equation:
  [ y = c_1 + c_2 e^{3x} ]
• Number of Constants: 2 (c1, c2)
• Derivatives:
  • First Derivative: [ y' = 3c_2 e^{3x} ]
  • Second Derivative: [ y'' = 9c_2 e^{3x} ]
• Differential Equation: [ y'' - 3y' = 0 ]

Example 4:

• Equation:
  [ y imes ext{sin}(x) - xy^2 = c ]
• Number of Constants: 1 (c)
• Derivatives:
  • After applying the product rule:
    [ y ext{cos}(x) + ext{sin}(x) y' - 2xyy' = 0 ]
• Simplified DE:
  [ y ext{cos}(x) - y^2 + ext{sin}(x) - 2xy = 0 ]

Example 5:

• Equation:
  [ x = c_1 ext{cos}( ext{ω}t) + c_2 ext{sin}( ext{ω}t) ]
• Number of Constants: 2 (c1, c2)
• Derivatives:
  • First Derivative: [ rac{dx}{dt} = - ext{ω}c_1 ext{sin}( ext{ω}t) + ext{ω}c_2 ext{cos}( ext{ω}t) ]
  • Second Derivative: [ rac{d^2x}{dt^2} = - ext{ω}^2(c_1 ext{cos}( ext{ω}t) + c_2 ext{sin}( ext{ω}t)) ]
• Differential Equation: [ rac{d^2x}{dt^2} + ext{ω}^2x = 0 ]

Example 6:

• Equation:
  [ x = a ext{sin}( ext{ω}t) + b ]
• Number of Constants: 2 (a, b)
• Derivatives:
  • First Derivative: [ rac{dx}{dt} = ext{ω}a ext{cos}( ext{ω}t) ]
  • Second Derivative: [ rac{d^2x}{dt^2} = - ext{ω}^2a ext{sin}( ext{ω}t) ]
• Differential Equation: [ rac{d^2x}{dt^2} + ext{ω}^2x = 0 ]

Example 7:

• Equation:
  [ y = c_1 e^{-2x} + c_2 e^{3x} ]
• Number of Constants: 2 (c1, c2)
• Derivatives:
  • First Derivative: [ y' = -2c_1 e^{-2x} + 3c_2 e^{3x} ]
  • Second Derivative: [ y'' = 4c_1 e^{-2x} + 9c_2 e^{3x} ]
• Differential Equation: [ y'' + 2y' = 0 ]

Conclusion

• Remember to determine the number of arbitrary constants to know how many derivatives to perform.
• The goal is to eliminate the arbitrary constants through differentiation and simplification.