Differential Equations ML Aggarwal ISC Class-12 Maths Solutions Ch-9

Differential Equations ML Aggarwal ISC Class-12 Maths Solutions Ch-9. Step by step Solutions of ML Aggarwal ISC Mathematics Class-12 for Exercise 9.1, Exe 9.2 And Exe 9.3, Exe 9.4, Exe 9.5, Exe 9.6, Exe 9.7, Exe 9.8, With Chapter Test Questions. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

Differential Equations ML Aggarwal ISC Class-12 Maths Solutions Ch-9

Board	ISC
Class	12
Subject	Mathematics
Chapter-9	Differential Equations
Session	2024-25
Topics	Solutions of ML Aggarwal

Differential Equations

Order of a Differential Equation: Order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation.
Note: Order of the differential equation, cannot be more than the number of arbitrary constants in the equation.

Degree of a Differential Equation: The highest exponent of the highest order derivative is called the degree of a differential equation provided exponent of each derivative and the unknown variable appearing in the differential equation is a non-negative integer.
Note
(i) Order and degree (if defined) of a differential equation are always positive integers.
(ii) The differential equation is a polynomial equation in derivatives.
(iii) If the given differential equation is not a polynomial equation in its derivatives, then its degree is not defined.

Formation of a Differential Equation:

To form a differential equation from a given relation, we use the following steps:

Step I: Write the given equation and see the number of arbitrary constants it has.
Step II: Differentiate the given equation with respect to the dependent variable n times, where n is the number of arbitrary constants in the given equation.
Step III: Eliminate all arbitrary constants from the equations formed after differentiating in step (II) and the given equation.
Step IV: The equation obtained without the arbitrary constants is the required differential equation.

Solution of the Differential Equation

A function of the form y = Φ(x) + C, which satisfies given differential equation, is called the solution of the differential equation.

General solution: The solution which contains as many arbitrary constants as the order of the differential equation, is called the general solution of the differential equation, i.e. if the solution of a differential equation of order n contains n arbitrary constants, then it is the general solution.

Particular solution: A solution obtained by giving particular values to arbitrary constants in the general solution of a differential equation, is called the particular solution.

General and Particular Solution of a Differential Equation-

The solution which contains arbitrary constants is called the general solution of the differential equation.

Whereas the solution free from arbitrary constants obtained from the general solution by giving particular values to the arbitrary constants is called a particular solution of the differential equation.

Formation of a Differential Equation whose General Solution is given

The order of a differential equation representing a family of curves is the same as the number of arbitrary constants present in the equation corresponding to the family of curves.