Differential equation means and example

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Differential equation means and example

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

Differential equation means and example

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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