Unit 10: Differential Equations - Explanation Of Exercise 10.1 (Mathematics-II For Class XII-Science Group)

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Mathematics II
For Class 12-(Science Group)
Unit 10: Differential Equations
Explanation Of Exercise 10.1

DIFFERENTIAL EQUATION:
It is defined as:
“An equation containing the derivatives of one or more dependent variables with respect to one independent variables.”
For a given function g, to find a function f such that

An above equation of the form (i) is known as differential equation.

ORDINARY DIFFERENTIAL EQUATION (DE):
“An equation involving derivatives ordinary derivatives of one or more dependent variables with respect to a single independent variable is called ordinary differential equation.”

ORDER OF THE DIFFERENTIAL EQUATION:
The order of a differential equation is the order of the highest derivative appearing in it.

DEGREE OF THE DIFFERENTIAL EQUATION:
“The degree of the differential equation is the degree of the highest order derivative occurring in it, after the equation has been expressed in a form free from radicals and non- integer powers of derivatives.”

SOLUTION OF DIFFERENTIAL EQUATION:
A solution of a differential equation is a relation between the variables free from derivatives, such that this relation and the derivatives obtained from it satisfies the given differential equation.

GENERAL AND PARTICULAR SOLUTION:
General Solution:
The general solution (complete solution) of a differential equation is the one in which the number of arbitrary constants is equal to the order of the differential equation.
Particular Solution:
A solution obtained from the general solution by giving particular values to the arbitrary constants is called particular solution.
For example, the differential equation ^d2y/_dx² + y = 0 has the general solutions y = A sin x + B cos x whose A & B are arbitrary constants.
When we assign fixed values to arbitrary constants according to given condition. For example, at y(0) = 1 and y'(0) = 2, we get A = 2 and B = 1, then the solution will be y = 2 sin x + cos x known as particular solution.

FORMATION OF DIFFERENTIAL EQUATION:
If the relation between the dependent variable and independent variable involves some arbitrary constants, we can form a differential equation by eliminating arbitrary constants from the relation by differentiating with respect to the independent variable successively as many times as the number of arbitrary constants.