NCERT Solutions For Class 12 Mathematics Chapter 9: Differential Equations

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The NCERT Solutions for class 12 mathematics chapter 9 Differential Equations are given in the article. Differential equation means the derivatives of a mathematical equation. The chapter Differential Equations belongs to the unit Calculus, that adds up to 35 marks of the total marks.

Chapter 9 of NCERT Solutions for Class 12 Maths covers the concepts of order and degree of differential equations, the method of solving a differential equation, their properties and much more.

Download: NCERT Solutions for Class 12 Mathematics Chapter 9 pdf

Class 12 Maths NCERT Solutions Chapter 9 Differential Equations

NCERT Solutions

Important Topics in Class 12 Mathematics Chapter 8 Applications of Integrals

Important concepts of Class 12 Maths covered in Chapter 9 Differential Equations of NCERT Solutions are:

Order of a differential equation

The order of a differential equation is defined to be of the highest order derivative it contains. Degree of a differential equation is defined as the power to which the highest order derivative is raised.

The equation (f‴)² + (f″)⁴ + f = x is an example of a second-degree, third-order differential equation.

How to Find Order of the Differential Equation?

The order of differential equation can be found by identifying the derivatives in the given expression of the differential equation. The different derivatives in a differential equation are as follows:

First Derivative:dy/dx or y'
Second Derivative: d²y/dx², or y''
Third Derivative: d³y/dx³, or y'''
nth derivative: dny/dxn, or y''''.....n times

Further, the highest derivative present in the differential equation defines the order of the differential equation, and the exponent of the highest derivative represents the degree of the differential equation.

Formation of a Differential Equation whose General Solution is given

For any given differential equation, the solution is of the form f(x,y,a¹,a², …….,an) = 0 where x and y are the variables and a¹ , a² ……. an are the arbitrary constants.

Methods of Solving First Order, First Degree Differential Equations

Different methods of solving first order, first degree differential equations are as follows:

Differential equations with variables separable
Homogeneous differential equations
Linear differential equations

Exercise Solutions of Class 12 Maths Chapter 9 Differential Equations

Also check Exercise Solutions of Class 12 Maths Chapter 9 Differential Equations

Also Read:

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CBSE CLASS XII Related Questions

1.
Smoking increases the risk of lung problems. A study revealed that 170 in 1000 males who smoke develop lung complications, while 120 out of 1000 females who smoke develop lung related problems. In a colony, 50 people were found to be smokers of which 30 are males. A person is selected at random from these 50 people and tested for lung related problems. Based on the given information answer the following questions:

(i) What is the probability that selected person is a female?
(ii) If a male person is selected, what is the probability that he will not be suffering from lung problems?
(iii)(a) A person selected at random is detected with lung complications. Find the probability that selected person is a female.
OR
(iii)(b) A person selected at random is not having lung problems. Find the probability that the person is a male.
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2.
Find the domain of \(p(x)=\sin^{-1}(1-2x^2)\). Hence, find the value of \(x\) for which \(p(x)=\frac{\pi}{6}\). Also, write the range of \(2p(x)+\frac{\pi}{2}\).
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3.
Find the sub–interval of \((0,\pi)\) in which the function \[ f(x)=\tan^{-1}(\sin x-\cos x) \] is increasing and decreasing.
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4.
Find the general solution of the differential equation \[ y\log y\,\frac{dx}{dy}+x=\frac{2}{y}. \]
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5.
A racing track is built around an elliptical ground whose equation is given by \[ 9x^2 + 16y^2 = 144 \] The width of the track is \(3\) m as shown. Based on the given information answer the following:

(i) Express \(y\) as a function of \(x\) from the given equation of ellipse.
(ii) Integrate the function obtained in (i) with respect to \(x\).
(iii)(a) Find the area of the region enclosed within the elliptical ground excluding the track using integration.
OR
(iii)(b) Write the coordinates of the points \(P\) and \(Q\) where the outer edge of the track cuts \(x\)-axis and \(y\)-axis in first quadrant and find the area of triangle formed by points \(P,O,Q\).
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6.
A rectangle of perimeter \(24\) cm is revolved along one of its sides to sweep out a cylinder of maximum volume. Find the dimensions of the rectangle.
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