CGPET 2018 Solved Paper

Given differential equation
x

dy

dx

=y−x‌tan(

y

x

)
⇒

dy

dx

=

y

x

−tan(

y

x

) ........(i)
which is homogeneous differential equation.
Now, put

y

x

=v
⇒y=vx⇒

dy

dx

=v+x

dv

dx

.........(ii)
From Eqs. (i) and (ii), we get
⇒v+x

dv

dx

=v−tan‌v
⇒x

dv

dx

=−tan‌v⇒

dv

tan‌v

=−

dx

x

⇒cot‌v‌d‌v=−

1

x

dx
On integrating both the sides, we get
log(sin‌v)=−log‌x+log‌C
⇒log(sin‌

y

x

)+log‌x=log‌C
⇒log(x.sin‌

y

x

)=log‌C
⇒x.sin(

y

x

)=C