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1. Differential Equations Ordinary and Partial differential equations

2. Content INVENTION OF DIFFERENTIAL EQUATION • INTRODUCTION TO DIFFERENTIAL EQUATION • TYPES OF DIFFERENTIAL EQUATION • BASIC CONCEPT OF ODE (ORDINARY DIFFERENTIAL EQUATION) • TYPES OF ODE • BASIC CONCEPT OF PDE (PARTIAL DIFFERENTIAL EQUATION) • TYPES OF PDE • APPLICATIONS OF DIFFERENTIAL EQUATIONS

3. Invention of Differential Equations :- Specifically, in 1693, both Leibniz & Newton finally, officially published & distributed solutions to their differential questions — marking 1693 as the inception for the differential equations as a distinct field in mathematics. The history of the subject of differential equations, in concise form, from a synopsis of the recent article "The History of Differential Equations, 1670-1950". "Differential equations began with Leibniz, the Bernoulli brothers, and others from the 1680s, not long after Newton's 'fluxional equations' in the 1670s."

4. Differential equation • A Differential Equation is an equation containing the derivative of one or more dependent variables with respect to one or more independent variables.

5. Types of Differential Equations ODE (ORDINARY DIFFERENTIAL EQUATION): PDE (PARTIAL DIFFERENTIAL EQUATION): An equation contains only ordinary derivates of one or more dependent variables of a single independent variable. An equation contains partial derivates of one or more dependent variables of two or more independent variables.

6. Types of ODE

7. First Order ODE

8. Second Order ODE

9. Higher Order ODE

10. For Example

11. APPLICATIONS OF ODE MODELLING WITH FIRST-ORDER EQUATIONS ■ Newton's Law of Cooling ■ Electrical Circuits MODELLING FREE MECHANICAL OSCILLATIONS ■ No Damping ■ Light Damping ■ Heavy Damping MODELLING FORCED MECHANICAL OSCILLATIONS COMPUTER EXERCISE OR ACTIVITY

12. Types of PDE Example of linear PDE: 2 Uxx+1 Uxt+3 u₁₁ +4 ux + cos(2t) = 0 2xx-34, +4 ux = 0 Examples of Nonlinear PDE 24xx+(x)+3 4 = 0 √xx+2 Ux+3u, = 0 XX 2 Uxx+2 4x4, +3и, = 0 A PDE is linear if it is linear in the unknown function and its derivatives