An equation containing **at least one** derivative of a dependent variable with respect to an independent variable** Differential** equation **means and example** such that

**y d y/d x +2x = 0 …………………………………….a**

**x d ²y / d x² + d y / d x – 2 x = 0………………………b**

is called **Equation (a) and Equation (b)** a differential equation.

Derivatives may be of first or higher orders. A differential equation containing only the derivative of first order can be written in terms of differential.

So we can write equation** (a) y d y +2x d x = 0**

But equation (b) cannot be written in the form of differential.

**Order of differential equation:**

The highest derivative in the equation is called order of a differential equation.

**First order differential equation:**

The form of the Equation (a) is called FIRST order differential Equation.

**Second order differential equation:**

The form of the Equation (b) is called SECOND order differential Equation

**Differential equation means and example of first order**:

EXAMPLE:

Consider the equation

**y = A x ² +4……………………… A**

Where A is the real constant

Differentiating Equation A with respect to x gives

**d y / d x = 2 A x…………………… B**

Equation A can be written as** A = y – 4 / x ²** this value putting in Equation B.

**d. y / d x = 2 (y – 4 / x ²) x**

**⇒ x d y / d x = 2 y – 8**

**⇒ 2 y – x d y / d x = 8…………………….. C**

Substituting the value of y and its derivative in C. we see that it is satisfied.

**2 (Ax ² + 4) – x (2 A x)**

**= 2 A x ² +8 – 2 A x ²**

**= 8**

Which shows that Equation A is a solution of Equation C.

Giving a particular value to A, say A = – 1 we get

y = – x ² + 4

**WE CHECK EQUATION C**

putting **y = – x ² + 4** and its derivative

**d y / d x = -2 x**

so** y = – x ²+ 4** is also a derivative of C.

**For different value of A on differential equation C:**

By putting different value of A. we get the **different parabola** with vertex at (0, 4) and the Axis along the y-axis. We have drawn two members of parabolas

**y = A x ² + 4 for A = – 1, 1**

All solution obtained from Equation A by putting different value of A, are called the particular solution of Equation C. While the solution C Itself is called the general solution of C.

A solution of a differential equation is a relation between the variable (not involving the derivative)Which satisfied the differential Equation.

We shall solve a differential Equation of first order with variables separable in the form.

**d. y / d x = f (x) / g (y) or d y / d x = g(x) / f(x)**

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