Consider the following homogeneous differential equation. x dx + (y − 2x) dy = 0 Use the substitution x = vy to write the given differential equation in terms of only y and v.

Question

Consider the following homogeneous differential equation.

x dx + (y − 2x) dy = 0

Use the substitution x = vy

 to write the given differential equation in terms of only y and v.

Answer 1

For x = vy, obtain y = x/v.

Also, dx/dy or d(vy)/dy = v(dy/dy) + (dv/dy)y

or v + y(dv/dy) which gives dx as vdy + ydv.

Then xdx + (y − 2x)dy = 0 in terms of v & y only

is written as vy(vdy + ydv) + (y − 2vy)dy = 0.

Further development gives v²ydy + vy²dv +ydy − 2vydy = 0

and then v²dy + vydv +dy − 2vdy = 0.

Rewrite this last as v²dy − 2vdy +dy + vydv = 0

or (v² − 2v + 1)dy = -vydv.

Separation of variables gives dy/y = -v dv/(v − 1)².

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The equation above can be taken further to

dy/y = -[1/(v − 1) + 1/(v − 1)²]dv which integrates to

ln |y| = -ln |v − 1| + (v − 1)^-1 + C.

For v equal to x/y, establish ln |y| = -ln |(x − y)/y| + ((x − y)/y)^-1 + C,

which goes to ln |x − y| = y/(x − y) + C.

Answer 2

xdx/dy + y = 2x; x = vy; dx/dy = v + ydv/dy.

vydx/dy + y = 2vy, reducing by y we get: dx/dy = 2y - 1