Lecture 1 [MATH 129]

 Lecture-1

Differential Equation

  Questions: Define the followings:

    (a) Differential Equation
    (b) Ordinary Differential Equation
    (c) Partial Differential Equation
    (d) Order of a Differential Equation
    (e) Degree of a Differential Equation

    Answer:

Differential equation: An equation involving independent variable, dependent variable and differential co-efficient of dependent variable with respect to independent variable is called a differential equation.

That is, the variables may or may not lie in differential equation but the differential co-efficient must be lie in the differential equation.

x is called independent variable and y is called dependent variable. Derivatives of y with respect to x is denoted by $\displaystyle \frac{dy}{dx}$  

 Example:

$$\begin{align*} (a) ~ &\frac{dy}{dx} + (a-y^2)\tan x = 0 \\ (b) ~ &\frac{dy}{dx}=1 \\ (c) ~ &x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}= nz \end{align*}$$

There are two types of differential equations:

1. Ordinary Differential equation
2. Partial differential equation

    Ordinary differential equation: An equation involving one independent variable one dependent variable and derivative of a dependent variable with respect to a single independent variable is called an ordinary differential equation.

    Example:

$$\begin{align*} (1) ~ &\frac{d^2y}{dx^2} -5 \frac{dy}{dx}+6y = 0 \\ (2) ~ & x^2\frac{d^2y}{dx^2}+2x\frac{dy}{dx}+y=0 \\ \end{align*}$$

    Partial differential equation: An equation involving one dependent variable, two or more independent variables and partial differential co-efficient of dependent variable with respect to more than one independent variable is called partial differential equation.

    Example:

$$\begin{align*} (1) ~ &x\frac{\partial z}{\partial x}+y \frac{\partial z}{\partial y}=32 \\ (2) ~ &\frac{\partial^2 z}{\partial x^2}+\frac{\partial^2 z}{\partial y^2}=0 \end{align*}$$

    Order of a differential equation: The order of the highest order differential co-efficient involved in a differential equation is called the order of the differential equation.

    Example:

$$\begin{align*} (1) ~ &\frac{d^2y}{dx^2}+3\frac{dy}{dx}+2y=0 \\ (2) ~ &\frac{d^4y}{dx^4}+7 \frac{d^2y}{dx^2}+5y=\cos^2x \end{align*}$$

In the above, equation (1) is of the second-order and the equation (2) is of the fourth-order.

    Degree of a differential equation: the degree or the power of the highest order derivative of a differential equation is called the degree of a differential equation when the equation has been made free from fractions.

    Example:

$$\begin{align*} (1) ~ &\frac{d^3y}{dx^3}+3\left((\frac{dy}{dx}\right)^4+y=0 \\ (2) ~ &\frac{d^2y}{dx^2}=k\left[1+\left(\frac{dy}{dx}\right)^2\right]^\frac{5}{3} \end{align*}$$

Since in the equation (1), the power of the highest third-order derivative is one, so the degree of the equation is one. If so the power of $\displaystyle\frac{dy}{dx}$ is 4 but $\frac{dy}{dx}$ is not the highest order differential co-efficient. So the degree of the equation is not 4. Hence the degree of the equation is one.

If eliminating the fractional power of equation (2), Then the equation is of the form 

$$\left(\frac{d^2y}{dx^2}\right)^3=k^3\left[1+\left(\frac{dy}{dx}\right)^2\right]^5$$

In this case, the power of the highest second-order derivative is 3. Hence the degree of the equation is 3.