NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations Ex 9.3

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Board	CBSE
Textbook	NCERT
Class	Class 12
Subject	Maths
Chapter	Chapter 9
Chapter Name	Differential Equations
Exercise	Ex 9.3
Number of Questions Solved	12
Category	NCERT Solutions

NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations Ex 9.3

In each of the following, Q. 1 to 5 form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

Ex 9.3 Class 12 Maths Question 1.
$frac { x }{ a } +frac { y }{ b } =1$
Solution:
Given that $frac { x }{ a } +frac { y }{ b } =1$ …(i)
differentiating (i) w.r.t x, we get
$frac { 1 }{ a } +frac { 1 }{ b } { y }^{ I }=0$ …(ii)
again differentiating w.r.t x, we get
$frac { 1 }{ b } { y }^{ II }=0Rightarrow { y }^{ II }=0$
which is the required differential equation

Ex 9.3 Class 12 Maths Question 2.
y² = a(b² – x²)
Solution:
given that
y² = a(b² – x²)…(i)
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 2

Ex 9.3 Class 12 Maths Question 3.
y = ae^3x+be^-2x
Solution:
Given that
y = ae^3x+be^-2x …(i)
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 3

Ex 9.3 Class 12 Maths Question 4.
y = e^2x (a+bx)
Solution:
y = e^2x (a+bx)
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 4

Ex 9.3 Class 12 Maths Question 5.
y = e^x(a cosx+b sinx)
Solution:
The curve y = e^x(a cosx+b sinx) …(i)
differentiating w.r.t x
byjus class 12 maths Chapter 9 Differential Equations 5

Ex 9.3 Class 12 Maths Question 6.
Form the differential equation of the family of circles touching the y axis at origin
Solution:
The equation of the circle with centre (a, 0) and radius a, which touches y- axis at origin
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 6

Ex 9.3 Class 12 Maths Question 7.
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Solution:
The equation of parabola having vertex at the origin and axis along positive y-axis is
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 7

Ex 9.3 Class 12 Maths Question 8.
Form the differential equation of family of ellipses having foci on y-axis and centre at origin.
Solution:
The equation of family ellipses having foci at y- axis is
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 8

Ex 9.3 Class 12 Maths Question 9.
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.
Solution:
Equation of the hyperbola is $frac { { x }^{ 2 } }{ { a }^{ 2 } } -frac { { y }^{ 2 } }{ { b }^{ 2 } } =1$
Differentiating both sides w.r.t x
byjus class 12 maths Chapter 9 Differential Equations 9
which is the req. differential eq. of the hyperbola.

Ex 9.3 Class 12 Maths Question 10.
Form the differential equation of the family of circles having centre on y-axis and radius 3 units
Solution:
Let centre be (0, a) and r = 3
Equation of circle is
x² + (y – a)² = 9 …(i)
Differentiating both sides, we get
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 10
which is required equation

Ex 9.3 Class 12 Maths Question 11.
Which of the following differential equation has $y={ c }_{ 1 }{ e }^{ x }+{ c }_{ 2 }{ e }^{ -x }$ as the general solution ?
(a) $frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +y=0$
(b) $frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -y=0$
(c) $frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +1=0$
(d) $frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -1=0$
Solution:
(b) $y={ c }_{ 1 }{ e }^{ x }+{ c }_{ 2 }{ e }^{ -x }Rightarrow frac { dy }{ dx } ={ c }_{ 1 }{ e }^{ x }-{ c }_{ 2 }{ e }^{ -x }$
$frac { { d }^{ 2 }y }{ { dx }^{ 2 } } ={ c }_{ 1 }{ e }^{ x }+{ c }_{ 2 }{ e }^{ -x }Rightarrow frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -y=0$

Ex 9.3 Class 12 Maths Question 12.
Which of the following differential equations has y = x as one of its particular solution ?
(a) $frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -{ x }^{ 2 }frac { dy }{ dx } +xy=x$
(b) $frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ x }frac { dy }{ dx } +xy=x$
(c) $frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -{ x }^{ 2 }frac { dy }{ dx } +xy=0$
(d) $frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ x }frac { dy }{ dx } +xy=0$
Solution:
(c) y = x
$frac { dy }{ dx } =1,frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =0$
$frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -{ x }^{ 2 }frac { dy }{ dx } +xy=0$

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NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations Ex 9.3

NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations Ex 9.3

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