Sample Material of Online Coaching For SSC CGL (Tier - 2) - Differential Equation


Sample Material SSC CGL TIER-2 Online Coaching


Numerical Aptitude (Chapter: Differential Equation)

Differential Equation

An equation containing and independent variable, dependent variable and differential coefficient of dependent variable with respect to independent variable is called a differential equation:
e.g. d3 y/dx3 -3(d2 y/dx2) +3dy/dx -y= sin 2x

Types of Differential Equations

(i) Ordinary differential equation (only one independent variable) e.g. dy/dx = y

(ii) Partial differential equation (two or more independent variables) e.g. x(dz/dx) + y(dz/dy)+z  = 0

Order of a Differential Equations

The order of a differential equation is the order of the highest order derivative appearing in the equation.
e.g. :d2 y/dx2 -3(dy/dx)  + 2y =e order = 2

Degree of a Differential Equation

The degree of a differential equation is the exponent of highest order derivative, when differential coefficient are made free from radicals and factions.

 [1+{dy/dx}2 ]3 = 9 {d2 y/dx2 }3, degree = 2

Primitive or  Solution of a Differential Equation

A primitive or solution of a differential equation is a functional relation between x and y which is free from derivatives and this relation on substitution satisfies the differential equation.

General Solution:

 It is the solution which contains the number of arbitrary constants equal to the order of the differential equation.

Particular Solution:

 A particular solution of differential equation is a solution obtained from the general solution by giving particular values to the arbitary constant.

Note:

 In exceptional case a relation containing n arbitrary constants may give rise to a differential equation of order less than n.
Formation of Differential Equation
Suppose
y = A e2x + Be–2x, eliminating arbitrary constant A and B i.e. and dy/dx = 2Ae2x - 2Be-2x

d2y/dx2 = 4Ae2x +4Be-2x =4y

i.e. d2y/dx2 -4y= 0

Solution of Ordinary Differential Equations of the First Order and First Degree An ordinary differential equation of the first order and the first degree is of the form
M+M (dx/dy )= 0
or Mdx + Ndy = 0
Where M and N are function of x and y or constant. The general solution of such equation will contain only one arbitary constant.

:: Home Assignment for Practice ::

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4. The slope of a curve at any point is the reciprocal of twice the ordinate at the point and it passes  through the point (4,3). Then the equation of the curve is:

(a) x2 = y + 5
(b) y2 = x – 5
(c) y2 = x + 5
(d) x2 = y – 5

 

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