# Thread: Solve the following ODE .. #8

1. ## Solve the following ODE .. #8

Hello

I have this problem

Problem:
Solve the following differential equation by using three different methods :

$\displaystyle (xy-1+y)dx+xdy=0$

Solution:

One method is finding the integrating factor which is $\displaystyle e^x$ which found by the formula $\displaystyle \displaystyle e^{\displaystyle \int \dfrac{\dfrac{\partial M}{\partial y}-\dfrac{\partial N}{\partial x}}{N} \, dx}$

what are the other 2 methods??

2. With some algebraic steps the DE becomes...

$\displaystyle \displaystyle y^{'} + y\ (1+\frac{1}{x^{2}}) = \frac{1}{x}$ (1)

... which is linear and can be solved with 'standard approach'...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. You could always do a series solution method!

4. If you had some I.C's then a numerical method like runge-kutta?

5. Originally Posted by chisigma
With some algebraic steps the DE becomes...

$\displaystyle \displaystyle y^{'} + y\ (1+\frac{1}{x^{2}}) = \frac{1}{x}$ (1)

... which is linear and can be solved with 'standard approach'...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
The 'standard approach' seems to lead to a solution like...

$\displaystyle \displaystyle y(x)= e^{-(x-\frac{1}{x})} \ \{\int \frac{e^{(x-\frac{1}{x})}}{x}\ dx + c \}$ (1)

... that contains a 'not very pleasant' integral ...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

6. The substitution $\displaystyle u=xy-1$ renders the equation separable:

$\displaystyle du=x\,dy+y\,dx,$ and hence if we have

$\displaystyle (xy-1)\,dx+y\,dx+x\,dy=0,$ then we arrive at

$\displaystyle u\,dx+du=0.$

7. Another one is by using inspection.

$\displaystyle (xy-1)dx+ydx+xdy=0$

$\displaystyle (xy-1)dx+d(xy)=0$

$\displaystyle \dfrac{d(xy)}{xy-1}+dx=0$ ... (1)

Actually, equation (1) is equivelant to the last equation in Ackbeet's replay.

8. Originally Posted by chisigma
With some algebraic steps the DE becomes...

$\displaystyle \displaystyle y^{'} + y\ (1+\frac{1}{x^{2}}) = \frac{1}{x}$ (1)

... which is linear and can be solved with 'standard approach'...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

You should get : $\displaystyle y'+\left(1+\dfrac{1}{x}\right) \cdot y = \dfrac{1}{x}$

9. Originally Posted by General

You should get : $\displaystyle y'+\left(1+\dfrac{1}{x}\right) \cdot y = \dfrac{1}{x}$
Of course!... and the 'standard approach' leads to the 'confortable' solution...

$\displaystyle \displaystyle y(x)= c\ \frac{e^{-x}}{x} + \frac{1}{x}$ (1)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

10. Thanks All.

11. You're welcome for my (this time, real!) contribution.

12. Thanks.