Question

The Integrating Factor of the D.E. 2xydy-(x² + y² + 1)dx = o is​

Answer 1

Correct option is

C

A family of rectangular hyperbiola with centre on x-axis

2xy  

dx

dy

​

=x  

2

+y  

2

+1

⇒2  

dx

dy

​

y−  

x

y  

2

 

​

=  

x

1

​

+x

Put v=y  

2

⇒  

dx

dv

​

=2y  

dx

dy

​

 

dx

dv

​

−  

x

v

​

=  

x

1

​

+x

Let μ=e  

∫−  

x

1

​

dx

=  

x

1

​

 

x

dx

dv

​

 

​

−  

x  

2

 

v

​

=−  

x

−  

x

1

​

−x

​

 

Put −  

x  

2

 

1

​

=  

dx

d

​

(  

x

1

​

)

x

dx

dv

​

 

​

+  

dx

d

​

(  

x

1

​

)v=−  

x

−  

x

1

​

−x

​

 

Using g  

dx

df

​

+f  

dx

dg

​

=  

dx

d

​

(fg)

dx

d

​

(  

x

v

​

)=−  

x

−  

x

1

​

−x

​

 

⇒∫  

dx

d

​

(  

x

v

​

)dx=∫−  

x

−  

x

1

​

−x

​

dx

⇒  

x

v

​

=−  

x

1

​

+x+c

⇒v=x  

2

+cx−1

⇒y  

2

=x  

2

+cx−1