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

Differential Equations ML Aggarwal ISC Class-12 Maths Understanding APC Solutions Chapter-9. Step by step Solutions of ML Aggarwal ISC Understanding APC Mathematics Class-12  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 APC Ch-9

Class:	12th
Subject:	Mathematics
Chapter  :	Ch-9 Differential Equations Page-533 to 652

Board	ISC
Writer	ML Aggarwal ISC Understanding Vol-II
Publications	APC Arya Publications

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-: Select Topics :-

Exe-9.1, 

Exe-9.2, 

Exe-9.3, 

Exe-9.4,

Exe-9.5,

Exe-9.6,

Exe-9.7,

Exe-9.8,

Chapter Test 

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

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.

Methods of Solving First Order, First Degree Differential Equations

There are three methods to solve the first-order, first-degree differential equations. They are:

• Differential equations with variables separable
• Homogeneous differential equations
• Linear differential equations

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Exe-9.1, 

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

Page 888, 889

Question 1: (dy/dx)- cos x = 0

Question 2: (dy/dx)² + (dy/dx) – sin²y = 0

Question 3:

………………….

………………….

Question 22: Find the product of the order and degree of the following different equations.

 

 

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Exe-9.2, 

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

Page 894, 895

Question 1: What is the number of ………………….. of order 4 ?

Question 2: What is the number of ………………….. of order of 3 ?

Question 3:  …………………

…………………..

Question 20: Show that y² = 4a (x + a) is a solutrion of the …………… = 2xy1.

 

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Exe-9.3, 

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

Page 905, 906

Question 1: From the differential equation of the family of ………….. constant.

Question 2 :  Find the ………… c are arbitrary constant.

Question 3:  ………………

………………….

Question 15: From the differential equations of simple ……….. are parameters.

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Exe-9.4, 

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

Page 925

Question 1: (i) (x² + 4)(dy/dx) = 1

(ii)……………

Question 2: (i) dy/dx = √4 – y²

(ii)…………

Question 3: …………………

…………………….

Question 27: Find the particular solution of ……….. when x = 0.

Exe-9.5, 

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

Page 931

Question 1: (i) (x + y)² dy/dx = 1

(ii)…………

Question 2: (i) (x – y)² dy/dx = a²

(ii)…………

Question 3:  ……………….

………………..

Question 6: Find the particular solution of the differential equation ……………… when x = -1.

 

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Exe-9.6, 

ML Aggarwal ISC Class-12 Maths APC Ch-9

Page 951

Question 1: When differential equation is called homogeneous.

Question 2: Which of the following differential equations are homogeneous ?

………………..

Question 3: …………………

………………….

Question 15: (i) Show that the following ………….. particular solutions :

……………..

 

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Exe-9.7, 

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

Page 979, 980

Question 1: which of the following equations are linear ………………. order in y  ?

………………..

Question 2: Which of the following  equations ……… in x ……………..

Question 3: …………………..

…………………..

Question 25: Find the particular solutions of the following differential equations ?

 

Exe-9.8, 

Differential Equations ISC Class-12 Maths APC Ch-9

Page 989

Question 1: The surface area of a balloon being ………… radius after t second.

Question 2: The surface area of a balloon ……………… radius after time t.

Question 3: …………..

………………………

Question 13:  Find the equation of a curve passing through the point ………………..  at the point by 5.

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Chapter Test

Differential Equations ML Aggarwal Class-12 Maths APC Ch-9

Page 990, 991

Question 1:  Show that ……………. = 0

Question 2: Verify that the function y = ………………………….. of the differential equation.

Question 3:……………………

……………………….

Question 18: Find the particular solutions of the following differential equations :

 

-: End of Differential Equations ML Aggarwal Vol-II ISC Class-12 Solutions:-

Return to :- ML Aggarwal ISC Class-12 APC Understanding Maths Solutions

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