Differential Equations Lecture Notes

Introduction

• Lecture focused on the recap of the chapter on Differential Equations.
• Key Topics: Order, Degree, Formation of Differential Equations, and Solving Methods.

1. Order and Degree of Differential Equations

1.1 Order

• Defined as the highest order of the derivative present in the equation.
  • Example:
    • ( \frac{dy}{dx} ) has order 1.
    • ( \frac{d^2y}{dx^2} ) has order 2.
    • ( \frac{d^3y}{dx^3} ) has order 3.
• To find the order, identify the highest derivative present.

1.2 Degree

• Defined as the power of the highest order derivative after expressing it as a polynomial in derivatives.
• Must check that all derivatives have whole number powers to confirm polynomial form.
  • Example: For ( \frac{d^2y}{dx^2} + a \cdot \frac{dy}{dx} = 0 ), degree is 1 (highest order derivative's power).
• If expressed as a polynomial, degrees can be assigned accordingly.

2. Formation of Differential Equations

• To form a differential equation from a family of curves:
  1. Differentiate the given equation as many times as the number of arbitrary constants.
  2. Eliminate arbitrary constants to derive the differential equation.

3. Methods of Solving Differential Equations

3.1 Separable Variables

• Differential equations can be expressed in the form: ( f(y) dy = g(x) dx ).
• Separate variables and integrate both sides.
  • Example:
    • ( \frac{dy}{dx} = f(x)g(y) ) can be separated as ( \frac{dy}{g(y)} = f(x) dx ).

3.2 Reducible to Separable Variables

• Some equations may not be separable but can be transformed:
  • Example: ( (x + a)^2 \frac{dy}{dx} = a^2 ) can be transformed for separation.

3.3 Linear Differential Equations

• Two types exist: in terms of ( x ) and in terms of ( y ).
• Form: ( \frac{dy}{dx} + P(x)y = Q(x) \)
• Use integrating factors to solve.
• Example of solving:
  1. Identify the integrating factor.
  2. Multiply through by this factor.
  3. Integrate to find the solution.

4. Applications of Differential Equations

• Applications in kinetics, physical chemistry, and orthogonal trajectories.
• Example of application: Population growth modeled by a proportionality constant based on the current population.

Conclusion

• The lecture covered essential concepts of Differential Equations, methods of solving them, and their applications.
• Suggested to practice problems for better understanding.