Differential Equations-help!!

Solve

(x+y)^2(x\frac{dy}{dx}+y)=xy(1+\frac{dy}{dx})

rearranging the formula would give,

x[(x+y)^2-y]\frac{dy}{dx}+y[(x+y)^2-x]=0[br] x[(x+y)^2-y]dy+y[(x+y)^2-x]dx=0

I have tried multiplying an integrating factor but I can't find one depending only on x or y. Is there a way of finding an integrating factor involving both x and y? or is there a substitution or a different method for these type of equations?

Thanks in advance for any help

Reply 1

ghostwalker

17

Original post by r2enigma

...

Differential equations is not my best subject area, but if you divide both sides of the original equation by the terms that don't involve the derivative, then you may recognize the form of each side and be able to integrate directly.

(edited 13 years ago)

Reply 2

jonathan3909

11

Original post by r2enigma

Solve

(x+y)^2(x\frac{dy}{dx}+y)=xy(1+\frac{dy}{dx})

rearranging the formula would give,

x[(x+y)^2-y]\frac{dy}{dx}+y[(x+y)^2-x]=0[br] x[(x+y)^2-y]dy+y[(x+y)^2-x]dx=0

I have tried multiplying an integrating factor but I can't find one depending only on x or y. Is there a way of finding an integrating factor involving both x and y? or is there a substitution or a different method for these type of equations?

Thanks in advance for any help

Use separating the variables.

Reply 3

r2enigma

OP

Original post by jonathan3909

Use separating the variables.

thanks for your reply. How do I separate x and y variables into two sides? Any hints?

Reply 4

jonathan3909

11

Original post by r2enigma

thanks for your reply. How do I separate x and y variables into two sides? Any hints?

The hint is in your above post :rolleyes:

Reply 5

jonathan3909

11

Original post by r2enigma

thanks for your reply. How do I separate x and y variables into two sides? Any hints?

Ok I'll help.Expand the above brackets and multiply throughout first.

Reply 6

jonathan3909

11

x^{3} y' + yx^{2} + 2yx^{2}y' + 2xy^{2} + xy^{2}y' + y^{3} = xy + xyy' where

y' = \frac{dy}{dx}.

Now write the whole thing as :

\frac {dy}{dx} = .....

Reply 7

jonathan3909

11

x^{3} y' + 2yx^{2}y' + xy^{2}y' - xyy' = xy - 2xy^{2} - yx^{2} - y^{3}

\frac {dy}{dx} (x^{3} + 2yx^{2} + xy^{2} -xy) = xy - 2xy^{2} - yx^{2} - y^{3}

Reply 8

jonathan3909

11

And put the y variables on dy side and x variables on dx side and then integrate to get y=....

Reply 9

DFranklin

18

Original post by jonathan3909

And put the y variables on dy side and x variables on dx sideHow?

Reply 10

DFranklin

18

Original post by ghostwalker

Differential equations is not my best subject area, but if you divide both sides of the original equation by the terms that don't involve the derivative, then you may recognize the form of each side and be able to integrate directly.