Introduction to Differential Equations

What is a Differential Equation?

• Definition: An equation involving the derivatives of a function.
• Example: Second derivative of y plus two times the first derivative of y equals three times y.
  • Can be written in different notations based on context:
    • Function notation: Second derivative of y with respect to x plus two times the first derivative equals three times y.
    • Leibniz notation: ( \frac{d^2y}{dx^2} + 2\frac{dy}{dx} = 3y )

Nature of Solutions

• Differential Equation Solutions:
  • The solution is a function or a class of functions.
  • Unlike algebraic equations, which have solutions as numbers or set of numbers.
• Example of Algebraic Equation:
  • ( x^2 + 3x + 2 = 0 ) has solutions x = -2, x = -1.

Verifying Solutions

• Example Differential Equation: ( \frac{d^2y}{dx^2} + 2\frac{dy}{dx} = 3y )
• Solution Examples:
  1. ( y_1(x) = e^{-3x} )
     • First derivative: ( -3e^{-3x} )
     • Second derivative: ( 9e^{-3x} )
     • Verification: ( 9e^{-3x} - 6e^{-3x} = 3e^{-3x} )
  2. ( y_2(x) = e^x )
     • First derivative: ( e^x )
     • Second derivative: ( e^x )
     • Verification: ( e^x + 2e^x = 3e^x )

Key Takeaways

• Multiple Solutions: Differential equations can have multiple solutions that form a class of functions.
• Understanding through Examples: Helps in understanding how solutions can satisfy given differential equations.

Next Steps

• Exploration of different classes of solutions.
• Techniques for solving and visualizing solutions.
• Developing a toolkit for deeper analysis of differential equations.