Differential Equations

Description:

    Differentiation and integration in solving first order, first-degree differential equations, and linear differential equations of order n; Laplace transforms in solving differential equations.

Overview:

    Differential equation, a mathematical statement containing one or more derivatives—that is, terms representing the rates of change of continuously varying quantities. Differential equations are very common in science and engineering, as well as in many other fields of quantitative study because what can be directly observed and measured for systems undergoing changes are their rates of change. The solution of a differential equation is, in general, an equation expressing the functional dependence of one variable upon one or more others; it ordinarily contains constant terms that are not present in the original differential equation. Another way of saying this is that the solution of a differential equation produces a function that can be used to predict the behavior of the original system, at least within certain constraints.

    Differential equations are classified into several broad categories, and these are in turn further divided into many subcategories. The most important categories are ordinary differential equations and partial differential equations. When the function involved in the equation depends on only a single variable, its derivatives are ordinary derivatives and the differential equation is classed as an ordinary differential equation. On the other hand, if the function depends on several independent variables so that its derivatives are partial derivatives, the differential equation is classed as a partial differential equation.

Read More:
(1) Differential equation | Britannica

(2) Differential Equations - Introduction - Math is Fun

(3) Differential Equations (Definition, Types, Order, Degree ...)

(4) S.O.S. Math - Differential Equations

(5) Applications of Differential Equations

Watch:
(1) What are Differential Equations and how do they work?

      by Sabine Hossenfelder [9:20] | YouTube

(2) This is why you're learning differential equations

      by Zach Star [18:35] | YouTube

(3) Solving Differential Equations in Filipino | Calculus | Paano?

      by Numberbender [8:22] | YouTube

(4) Differential equation introduction | First order differential equations | Khan Academy

      by Khan Academy [7:49] | YouTube

(5) Introduction to Differential Equations Order, Degree, Linearity (Tagalog/Filipino Math)

      by enginerdmath [15:46] | YouTube

Topics:

1. Basic Concepts of Differential Equations
        a. Definition and Classifications of Differential Equations (D.E.)
        b. Order Degree of a D.E. / Linearity
        c. Solution of a D.E. (General and Particular)

2. Solution of Some 1st Order, 1st Degree D.E.
        a. Variable Separable
        b. Exact Equation
        c. Linear Equation
        d. Substitution Methods

                i. Homogeneous Coefficients
                ii. Bernoulli's Equation

        e. Mixed Problems

        f. Introduction to Use of Computer in Solving D.E.

3. Applications of 1st Order D.E.
        a. Decomposition (Decay) / Growth
        b. Newton's Law of Cooling
        c. Mixing Problems (Non-Reacting Fluids)
        d. Electric Circuits

4. Linear D.E. of Order n
        a. Standard Form of a Linear D.E.
        b. Linear Independence of a Set of Functions
        c. Differential Operators
        d. Differential Operator Form of a Linear D.E.

        e. Principle of Superposition

5. Homogeneous Linear D.E. with Constant Coefficients
        a. General Solution
        b. Auxiliary Equation

        c. Initial and Boundary Value Problems

6. Non-Homogeneous D.E. with Constant-Coefficients
        a. Form of the General Solution
        b. Solution by Method of Undetermined Coefficients
        c. Solution by Variation of Parameters

        d. Mixed Problems

        e. Solution of Higher Order D.E. using Computer

7. Laplace Transforms of Functions

        a. Definition

        b. Transform of Elementary Functions

        c. Transform of e^at f(t) - Theorem

        d. Transform of t^n f(t) - Derivatives of Transforms

        e. Inverse Transform

        f. Laplace and Inverse Laplace Transform using Computer

        g. Transforms of Derivatives

        h. Initial Value Problems

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