General And Particular Solutions Of A Differential Equation

The correct option is(B): $(x%5E%7B2%7D-y%5E%7B2%7D)%20%2B2xy%20%5Cfrac%7Bdy%7D%7Bdx%7D%3D%200$ .

Let centre of circle on X-axis be $(h%2C%200)$
The radius of circle will be h

$%5Ctherefore$ The equation of circle having centre $(h%2C%200)$ and radius $h$ is
$(x-h)%5E%7B2%7D%20%2B%20(y-0)%5E%7B2%7D%20%3D%20h%5E%7B2%7D$
$%5CRightarrow%20x%5E%7B2%7D%2Bh%5E%7B2%7D-2hx%2By%5E%7B2%7D%20%3D%20h%5E%7B2%7D$
$%5CRightarrow%20x%5E%7B2%7D-2hx%2By%5E%7B2%7D%3D0%20%5C%2C%20%5Cdots(i)$
On differentiating both sides w.r.t x, we get
$2x-2h%2B2y%20%5Cfrac%7Bdy%7D%7Bdx%7D%3D0$
$%5CRightarrow%20h%3Dx%2By%20%5Cfrac%7Bdy%7D%7Bdx%7D$
On putting $h%20%3D%20x%2By%20%5Cfrac%7Bdy%7D%7Bdx%7D$ in E (i), we get
$x%5E%7B2%7D-2%5Cleft(x%2By%20%5Cfrac%7Bdy%7D%7Bdx%7D%5Cright)%20x%2By%5E%7B2%7D%3D0$
$%5CRightarrow%20-x%5E%7B2%7D%2By%5E%7B2%7D-2xy%20%5Cfrac%7Bdy%7D%7Bdx%7D%3D0$
$%5CRightarrow%20%5Cleft(x%5E%7B2%7D-y%5E%7B2%7D%5Cright)%2B2xy%20%5Cfrac%7Bdy%7D%7Bdx%7D%3D0$

We have,

%5Csin%20x%2B%5Csin%203%20x%2B%5Csin%205%20x%3D0

%5CRightarrow%20%5Csin%20x%2B%5Csin%205%20x%2B%5Csin%203%20x%3D0

%5CRightarrow%202%20%5Csin%203%20x%20%5Ccos%202%20x%2B%5Csin%203%20x%3D0

%5CRightarrow%20%5Csin%203%20x(2%20%5Ccos%202%20x%2B1)%3D0

%5CRightarrow%20%5Csin%203%20x%3D0

%5Ccos%202%20x%3D-%5Cfrac%7B1%7D%7B2%7D%3D%5Ccos%20%5Cfrac%7B2%20%5Cpi%7D%7B3%7D

%5CRightarrow%203%20x%3Dn%20%5Cpi%20%5Ctext%20%7B%20or%20%7D%202%20n%3D2%20n%20%5Cpi%20%5Cpm%20%5Cfrac%7B2%20%5Cpi%7D%7B3%7D

%5CRightarrow%20x%3D%5Cfrac%7Bn%20%5Cpi%7D%7B3%7D

x%3Dn%20%5Cpi%20%5Cpm%20%5Cfrac%7B%5Cpi%7D%7B3%7D

But, it is given that

x%20%5Cin%5Cleft%5B%5Cfrac%7B%5Cpi%7D%7B2%7D%2C%20%5Cfrac%7B3%20%5Cpi%7D%7B2%7D%5Cright%5D

%5Ctherefore%20x%3D%5Cfrac%7B2%20%5Cpi%7D%7B3%7D%2C%20%5Cpi%2C%20%5Cfrac%7B4%20%5Cpi%7D%7B3%7D

%5Ctherefore

Number of solutions is 3 .

General And Particular Solutions Of A Differential Equation

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

General And Particular Solutions Of A Differential Equation

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

Chapter Questions
2 MCQs