Question

solve homogeneous differential equation dy/dx=x+2y/2x+y​

Answer 1

Answer:

It is given that Here the differential equation is a homogeneous equation So the solution of the differential equation is Consider y = vx By differentiating w.r.t x By integrating both sides We get (y – x) = C (y + x)3Read more on Sarthaks.com - https://www.sarthaks.com/818627/in-differential-equations-show-that-it-is-homogeneous-and-solve-it-dy-dx-x-2y-2x-y-0

Answer 2

Answer:

The required solution of the given differential equation is  log( 1 + y/x ) = x + c . Where c is an arbitrary constant .

Step-by-step explanation:

The given differential equation is a homogenous differential equation. So we will solve it as follows :

the given differential equation is ,

dy/dx=x+2y/2x+y​

⇒dy/dx = x + y/x + y

let , y/x = v

⇒ y = vx

differentiating both sides we get ,

dy/dx = v + x( dv/dx )

then , dy/dx = x + y/x + y

⇒ v + x( dv/dx ) = x + v + vx

⇒ v + x( dv/dx ) = v + x( 1 + v )

⇒ x( dv/dx ) = x( 1 + v )   [ by left cancelation law ]

⇒ dv/dx = 1 + v  [ by left cancelation law ]

⇒dv/( 1 + v ) = dx

integrating both sides we get ,

⇒∫dv/( 1 + v ) = ∫dx

⇒ log( 1 + v ) = x + c

⇒log( 1 + y/x ) = x + c [ ∵ v = y/x ]

∴ the solution is ,

log( 1 + y/x ) = x + c , where c is an arbitrary constant.