solve differential equation HELP pls

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March 8th 2009, 11:25 AM
manalive04

solve differential equation HELP pls

dy/dx= (2x-y)/(x+y) all subject to y(1) = 1

I am not sure where to take this. I divided everything by x and made the subsequent y/x = V and then worked from there. please can you help
March 8th 2009, 11:49 AM
HallsofIvy

Quote:

Originally Posted by manalive04

dy/dx= (2x-y)/(x+y) all subject to y(1) = 1

I am not sure where to take this. I divided everything by x and made the subsequent y/x = V and then worked from there. please can you help

That sounds like a very good idea! What did you get?
March 8th 2009, 11:59 AM
manalive04

i got an improper fraction which i am not sure what to do with

x(dv/dx) = (2 - 2v - v^2)/1+v

then took everything to the other side ==>

(1+v)/(2-2v-v^2)dv = dx/x

then to intergrate both sides

but how do you do the LHS? it is topheavy?
March 8th 2009, 12:06 PM
skeeter

Quote:

Originally Posted by manalive04

i got an improper fraction which i am not sure what to do with

x(dv/dx) = (2 - 2v - v^2)/1+v

then took everything to the other side ==>

(1+v)/(2-2v-v^2)dv = dx/x

then to intergrate both sides

but how do you do the LHS? it is topheavy?

$\frac{1+v}{2-2v-v^2} \, dv = \frac{1}{x} \, dx$

$\frac{-2 - 2v}{2-2v-v^2} \, dv = -\frac{2}{x} \, dx$

can you integrate the LHS now?
March 8th 2009, 12:12 PM
manalive04

Quote:

Originally Posted by skeeter

$\frac{1+v}{2-2v-v^2} \, dv = \frac{1}{x} \, dx$

$\frac{-2 - 2v}{2-2v-v^2} \, dv = -\frac{2}{x} \, dx$

can you integrate the LHS now?

can you plese show me how im sorry im not confident here
March 8th 2009, 12:19 PM
skeeter

hint: $\int \frac{u'}{u} \, dv$
March 8th 2009, 12:22 PM
manalive04

Quote:

Originally Posted by skeeter

hint: $\int \frac{u'}{u} \, dv$

what is u'

im sorry i dont understand
March 8th 2009, 12:37 PM
e^(i*pi)

Quote:

Originally Posted by manalive04

what is u'

im sorry i dont understand

u is just a variable in this case.

$\int{\frac{u'}{u}}$ is a special case of integral and it equals $ln|x| + C$

it is saying that differentiating $2-2v-v^2$ (the denominator) gives -2-2v which is the numerator. The poster said that u = 2-2v-v^2 and thus u' = -2-2v
March 8th 2009, 12:44 PM
manalive04

Quote:

Originally Posted by e^(i*pi)

u is just a variable in this case.

$\int{\frac{u'}{u}}$ is a special case of integral and it equals $ln|x| + C$

it is saying that differentiating $2-2v-v^2$ (the denominator) gives -2-2v which is the numerator. The poster said that u = 2-2v-v^2 and thus u' = -2-2v

so what does this imply???

im so confused can u please make this clearer im sorry
March 8th 2009, 01:51 PM
skeeter

$\frac{1+v}{2-2v-v^2} \, dv = \frac{1}{x} \, dx$

$\frac{-2 - 2v}{2-2v-v^2} \, dv = -\frac{2}{x} \, dx$

$\ln|2 - 2v - v^2| = -2\ln|x| + C$