making equation exact

**duaneg37** · February 9th 2011, 12:51 PM

Solve $(x^{2}+y^{2}-5)dx=(y+xy)dy$
$y(0)=1$

I have tried finding a factor to make this equation exact, and I get $(x+1)^{-3}$ . I then integrate by parts with respect to x and get $\,{\frac {-{x}^{2}-{y}^{2}+5}{2 \left( x+1 \right) ^{2}}}+\ln \left( x+1 \right) + \left( x+1 \right) ^{-1}+\rho \left( y \right) = {\frac {{x}^{2}+{y}^{2}-5}{ \left( x+1 \right) ^{3}}} $
I feel like I'm not getting anywhere because nothing cancels like it should. Can someone please help me? Thanks!

**General** · February 9th 2011, 02:23 PM

Rewrite your differential as
$(x^{2}+y^{2}-5)dx-y(1+x)dy=0$

Multiplying by the integrating factor

$ \displaystyle \dfrac{x^2+y^2-5}{(x+1)^3} dx + \left( \dfrac{-y}{(x+1)^2} \right) dy = 0 $

The last equation is exact. Its solution of the form $F(x,y)=C$ where :

$F_x=\dfrac{x^2+y^2-5}{(x+1)^3}$ ... (1)

$F_y=\dfrac{-y}{(x+1)^2}$ ... (2)

I think you integrate (1) with respect to x. This is too complicated .

Try to integrate (2) with respect to y and put p(x) as the constant of integration.

**duaneg37** · February 9th 2011, 05:47 PM

$\frac{\partial{f}}{\partial{y}}={\frac {-{y}{\it }}{ \left( x+1 \right) ^{2}}}dy$ Now I integrate w.r.t.y

$f( x,y) ={\frac {-{y}^{2}}{ \left2( x+1 \right) ^{2}} }+p \left( x \right)$ Now I take partial w.r.t.x

$\frac{\partial{f}}{\partial{x}}={\frac {-{y}^{2}}{4(x+1)}}+\mbox p' \left( x \right) ={\frac { \left( {x}^{2}+{y}^{2}-5 \right) {\it }}{ \left( x+1 \right) ^{3}}}$ I thought things would cancel if it was exact. Can anyone tell me what I'm doing wrong? Thanks!

**duaneg37** · February 9th 2011, 07:29 PM

${{\rm e}^{\int \!{\frac {M_{{y}}-N_{{x}}}{N}}{dx}}}$

This is how I get $(x+1)^{-3}$

How do you get x+1?

**topsquark** · February 9th 2011, 08:01 PM

Originally Posted by duaneg37

${{\rm e}^{\int \!{\frac {M_{{y}}-N_{{x}}}{N}}{dx}}}$

This is how I get $(x+1)^{-3}$

How do you get x+1?

Okay, I see my mistake. Hang on.

-Dan

**topsquark** · February 9th 2011, 08:31 PM

Originally Posted by General

Rewrite your differential as
$(x^{2}+y^{2}-5)dx-y(1+x)dy=0$

Multiplying by the integrating factor

$ \displaystyle \dfrac{x^2+y^2-5}{(x+1)^3} dx + \left( \dfrac{-y}{(x+1)^2} \right) dy = 0 $

The last equation is exact. Its solution of the form $F(x,y)=C$ where :

$F_x=\dfrac{x^2+y^2-5}{(x+1)^3}$ ... (1)

$F_y=\dfrac{-y}{(x+1)^2}$ ... (2)

I think you integrate (1) with respect to x. This is too complicated .

Try to integrate (2) with respect to y and put p(x) as the constant of integration.

Okay, now that I've fixed my little problem with the minus signs...
(1) isn't difficult to integrate, but it is tedious. You should get
$F(x, y) = ln|x + 1| + \frac{4x -y^2 + 8}{2(x + 1)^2} + \phi (y)$

What I've always done from here is to integrate (2). This is an easy one. Then just match up common terms. It saves from having to work with the derivative of what you called p(x).

-Dan

**duaneg37** · February 9th 2011, 09:08 PM

$f(x,y)=\ln \left( x+1 \right) +\,{\frac {4\,x-{y}^{2}+8}{ 2\left( x+1 \right) ^{2}}}-\frac{7}{2}=1$

Does this look right? What technique did you use to integrate $F_{{x}}$ ?

**topsquark** · February 9th 2011, 11:06 PM

$F(x, y) = \int \frac{x^2 + y^2 - 5}{(x + 1)^3} dx + \phi(y)$

So we need to evaluate the integral
$\int \frac{x^2 + y^2 - 5}{(x + 1)^3} dx$

$= \int \frac{x^2}{(x + 1)^3} dx + \int \frac{y^2 - 5}{(x + 1)^3}~dx$

The second integral we can evaluate directly:
$= \int \frac{x^2}{(x + 1)^3} dx - \frac{y^2 - 5}{2(x + 1)^2}$

For the first integral use a = x + 1:

$= \int \frac{(a - 1)^2}{a^3}da - \frac{y^2 - 5}{2(x + 1)^2}$

$= \int \left ( \frac{1}{a} - \frac{2}{a^2} + \frac{1}{a^3} \right ) da - \frac{y^2 - 5}{2(x + 1)^2}$

You can fill in the rest from there.

-Dan

**topsquark** · February 9th 2011, 11:14 PM

Originally Posted by duaneg37

$f(x,y)=\ln \left( x+1 \right) +\,{\frac {4\,x-{y}^{2}+8}{ 2\left( x+1 \right) ^{2}}}-\frac{7}{2}=1$

Does this look right? What technique did you use to integrate $F_{{x}}$ ?

What did you get when you integrated that second function?
$F(x, y) = -\int \dfrac{y}{(x + 1)^2}dy + \omega (x) = -\dfrac{1}{(x + 1)^2} \int y~dy + \omega (x)$

$= -\dfrac{y^2}{2(x + 1)^2} + \omega (x)$

Now just compare the two forms.

By the way, please note the distinction between $ln(x + 1)$ and the correct $ln|x + 1|$

-Dan

**General** · February 10th 2011, 05:40 AM

Originally Posted by duaneg37

$\frac{\partial{f}}{\partial{y}}={\frac {-{y}{\it }}{ \left( x+1 \right) ^{2}}}dy$ Now I integrate w.r.t.y

$f( x,y) ={\frac {-{y}^{2}}{ \left2( x+1 \right) ^{2}} }+p \left( x \right)$ Now I take partial w.r.t.x

$\frac{\partial{f}}{\partial{x}}={\frac {-{y}^{2}}{4(x+1)}}+\mbox p' \left( x \right) ={\frac { \left( {x}^{2}+{y}^{2}-5 \right) {\it }}{ \left( x+1 \right) ^{3}}}$ I thought things would cancel if it was exact. Can anyone tell me what I'm doing wrong? Thanks!

Your $\displaystyle \frac{\partial{f}}{\partial{x}}$ is wrong.

You should get : $\displaystyle \dfrac{y^2}{(x+1)^3}+p'(x)$ ... (1)

And , from the original equation since M= $F_x$ we have $\displaystyle \frac{\partial{f}}{\partial{x}}=\dfrac{x^2+y^2-5}{(x+1)^3}$

Or : $\displaystyle \frac{\partial{f}}{\partial{x}}=\dfrac{y^2}{(x+1)^ 3}+\dfrac{x^2-5}{(x+1)^3}$ ... (2)

Comparing (1) & (2) gives : $\displaystyle p'(x)=\dfrac{x^2-5}{(x+1)^3}$

I think you can continue ...

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