Introduction to Differential Equations

Definition

• Differential Equation: An equation that relates a function with its derivatives.
• Example given: Second derivative of y plus two times the first derivative of y equals three times y.

Notation

• Function Notation: ( f''(x) + 2f'(x) = 3f(x) )
• Leibniz Notation: ( \frac{d^2y}{dx^2} + 2\frac{dy}{dx} = 3y )

Solutions to Differential Equations

• Solution: A function, or a class of functions, that satisfies the differential equation.
• Contrast with Algebraic Equations: Solutions to algebraic equations are usually numbers or a set of numbers.
  • Example of algebraic equation: ( x^2 + 3x + 2 = 0 ); solutions are ( x = -2 ) or ( x = -1 ).

Example Solutions

1. First Solution (( y_1 )):

   • ( y_1(x) = e^{-3x} )
   • Derivatives:
     • ( y_1'(x) = -3e^{-3x} )
     • ( y_1''(x) = 9e^{-3x} )
   • Verification:
     • Substitute back into the differential equation: ( 9e^{-3x} - 6e^{-3x} = 3e^{-3x} )
     • Confirms that ( y_1 ) is a solution.

2. Second Solution (( y_2 )):

   • ( y_2(x) = e^x )
   • Derivatives:
     • ( y_2'(x) = e^x )
     • ( y_2''(x) = e^x )
   • Verification:
     • Substitute back into the differential equation: ( e^x + 2e^x = 3e^x )
     • Confirms that ( y_2 ) is also a solution.

Conclusion

• Multiple solutions exist for differential equations.
• Upcoming topics:
  • Exploring more solutions and classes of solutions.
  • Techniques for solving differential equations.
  • Visualization of solutions and further exploration.