NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations Ex 9.5

NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations Ex 9.5 are part of NCERT Solutions for Class 12 Maths. Here we have given NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations Ex 9.5.

Board	CBSE
Textbook	NCERT
Class	Class 12
Subject	Maths
Chapter	Chapter 9
Chapter Name	Differential Equations
Exercise	Ex 9.5
Number of Questions Solved	17
Category	NCERT Solutions

NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations Ex 9.5

Show that the given differential equation is homogeneous and solve each of them in Questions 1 to 10

Question 1.
(x²+xy)dy = (x²+y²)dx
Solution:
(x²+xy)dy = (x²+y²)dx
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 1

Question 2.
${ y }^{ I }=\frac { x+y }{ x }$
Solution:
${ y }^{ I }=\frac { x+y }{ x }$
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 2

Question 3.
(x-y)dy-(x+y)dx=0
Solution:
$\frac { dy }{ dx } =\frac { x+y }{ x-y } =\frac { 1+\frac { y }{ x } }{ 1-\frac { y }{ x } }$
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 3

Question 4.
(x²-y²)dx+2xy dy=0
Solution:
$\frac { dy }{ dx } =\frac { { y }^{ 2 }-{ x }^{ 2 } }{ 2xy }$
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 4

Question 5.
${ x }^{ 2 }\frac { dy }{ dx } ={ x }^{ 2 }-{ 2y }^{ 2 }+xy$
Solution:
$\frac { dy }{ dx } =1-2{ \left( \frac { y }{ x } \right) }^{ 2 }+\frac { y }{ x }$
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 5

Question 6.
$xdy-ydx=\sqrt { { x }^{ 2 }+{ y }^{ 2 } } dx$
Solution:
$xdy-ydx=\sqrt { { x }^{ 2 }+{ y }^{ 2 } } dx$
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 6

Question 7.
$\left\{ xcos\left( \frac { y }{ x } \right) +ysin\left( \frac { y }{ x } \right) \right\} ydx=\left\{ ysin\left( \frac { y }{ x } \right) -xcos\left( \frac { y }{ x } \right) \right\} xdy$
Solution:
$\left\{ xcos\left( \frac { y }{ x } \right) +ysin\left( \frac { y }{ x } \right) \right\} ydx=\left\{ ysin\left( \frac { y }{ x } \right) -xcos\left( \frac { y }{ x } \right) \right\} xdy$
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 7

Question 8.
$x\frac { dy }{ dx } -y+xsin\left( \frac { y }{ x } \right) =0$
Solution:
$x\frac { dy }{ dx } -y+xsin\left( \frac { y }{ x } \right) =0\Rightarrow \frac { dy }{ dx } =\frac { y }{ x } -sin\frac { y }{ x }$
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 8

Question 9.
$ydx+xlog\left( \frac { y }{ x } \right) dy-2xdy=0$
Solution:
$\frac { dy }{ dx } =\frac { y }{ 2x-xlog\frac { y }{ x } } =\frac { \frac { y }{ x } }{ 2-log\frac { y }{ x } }$
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 9

Question 10.
$\left( { 1+e }^{ \frac { x }{ y } } \right) dx+{ e }^{ \frac { x }{ y } }\left( 1-\frac { x }{ y } \right) dy=0$
Solution:
$\frac { dx }{ dy } =-\frac { { e }^{ \frac { x }{ y } }\left( 1-\frac { x }{ y } \right) }{ { 1+e }^{ \frac { x }{ y } } } =\frac { \left( \frac { x }{ y } -1 \right) { e }^{ \frac { x }{ y } } }{ { 1+e }^{ \frac { x }{ y } } } =f(x,y)$
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 10

For each of the following differential equation in Q 11 to 15 find the particular solution satisfying the given condition:

Question 11.
(x + y) dy+(x – y)dx = 0,y = 1 when x = 1
Solution:
given
(x + y) dy+(x – y)dx = 0
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 11

Question 12.
x²dy+(xy+y²)dx=0, y=1 when x=1
Solution:
$\frac { dy }{ dx } =\frac { xy+{ y }^{ 2 } }{ { x }^{ 2 } } =f(x,y)$
f(x,y) is homogeneous
∴ put y = vx
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 12

Question 13.
$\left( x{ sin }^{ 2 }\frac { y }{ x } -y \right) dx+xdy=0,y=\frac { \pi }{ 4 } ,when\quad x=1$
Solution:
$\left( x{ sin }^{ 2 }\frac { y }{ x } -y \right) dx+xdy=0$
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 13

Question 14.
$\frac { dy }{ dx } -\frac { y }{ x } +cosec\left( \frac { y }{ x } \right) =0,y=0\quad when\quad x=1$
Solution:
$\frac { dy }{ dx } -\frac { y }{ x } +cosec\left( \frac { y }{ x } \right) =0$
which is a homogeneous differential equation
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 14

Question 15.
$2xy-{ y }^{ 2 }-{ 2x }^{ 2 }\frac { dy }{ dx } =0,y=2,when\quad x=1$
Solution:
$\frac { dy }{ dx } =\frac { y }{ x } +\frac { 1 }{ 2 } { \left( \frac { y }{ x } \right) }^{ 2 }$ …(i)
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations 15

Question 16.
A homogeneous equation of the form $\frac { dx }{ dy } =h\left( \frac { x }{ y } \right)$ can be solved by making the substitution,
(a) y=vx
(b) v=yx
(c) x=vy
(d) x=v
Solution:
(c) option x = vy

Question 17.
Which of the following is a homogeneous differential equation?
(a) (a) (4x + 6y + 5)dy-(3y + 2x + 4)dx = 0
(b) $(xy)dx-({ x }^{ 3 }+{ y }^{ 3 })dy$
(c) $({ x }^{ 3 }+{ 2y }^{ 2 })dx+2xydy=0$
(d) ${ y }^{ 2 }dx+{ (x }^{ 2 }-xy-{ y }^{ 2 })dy=0$
Solution:
(d)

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NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations Ex 9.5

NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations Ex 9.5

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