Question

the given differential equation (2xy+y^2)dx+(x^2+2xy)dy=o is equation {exact or non-exact}​

Answer 1

Answer:

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Answer 2

Answer:

Step-by-step explanation:

Given, (x  

2

+y  

2

)dx−2xydy=0

⇒(x  

2

+y  

2

)dx=2xydy

⇒  

dx

dy

​  

=  

2xy

x  

2

+y  

2

 

​  

    .... (i)

Let y=vx

Thus,  

dx

dy

​  

=v+x  

dx

dv

​  

 

Thus, v+x  

dx

dv

​  

=  

2x(vx)

x  

2

+(vx)  

2

 

​  

 

⇒v+x  

dx

dv

​  

=  

2v

1+v  

2

 

​  

 

⇒x  

dx

dv

​  

=  

2v

1+v  

2

 

​  

−v

⇒x  

dx

dv

​  

=  

2v

1+v  

2

−2v  

2

 

​  

 

⇒x  

dx

dv

​  

=  

2v

1−v  

2

 

​  

 

⇒  

x

dx

​  

=  

1−v  

2

 

2v

​  

dv

⇒  

x

dx

​  

−  

1−v  

2

 

2v

​  

dv=0    .... (ii)

Integrating both sides, we have

logx+log(1−v  

2

)=logC

⇒logx(1−v  

2

)=logC

⇒x(1−v  

2

)=C

⇒x(1−  

x  

2

 

y  

2

 

​  

)=C

⇒x(  

x  

2

 

x  

2

−y  

2

 

​  

)=C

⇒x  

2

−y  

2

=Cx