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

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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
$\frac { 1 }{ b } { y }^{ II }=0\Rightarrow { y }^{ II }=0$
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
$\frac { dy }{ dx } =1,\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =0$
$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } -{ x }^{ 2 }\frac { dy }{ dx } +xy=0$

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

NCERT Solutions for Class 12 Maths Exercise 9.3 of Differential Equations

12 Maths Exercise 9.3 solutions
12 Maths Exercise 9.3 all answers
12 Maths Exercise 9.3 in English Medium