1. ## Ode

Has this old saw ever been answered?
Does the differential equation:
dy/dx=1/x+1/y
have a solution in terms of elementary functions?

2. I don't think so

This doesn't mean we can't solve it - even approximate the solution numerically.

As I sucked in Lebesgue spaces, I assume $\displaystyle y$ is smooth, and for the sake of well-posedness, $\displaystyle y(1)=a.$

Then $\displaystyle y'(1)=1+\frac{1}{a}, \ y''(1)=-1-\frac{1}{a^2}-\frac{1}{a^3}$ etc,

and a Taylor's expansion sais $\displaystyle y(x)=a+(x-1)y'(1)+\frac{(x-1)^2}{2}y''(1)+...$

so

$\displaystyle y(x)=a+(x-1)(1+\frac{1}{a})+...$

3. ## Ode

Thanks for responding. I appreciate your interest in this question.

Of course you can solve it numerically! But can the numerical solution be
shown to be an ELEMENTARY function? That's the question.

4. ...Well I already said i don't think so!

5. ## Ode

Yes you did! And thanks again!

6. I did try to solve this but alas I am having little success.

Rewrite,
$\displaystyle y'-y^{-1}=x^{-1}$
Note, this is a non-homogenous.
Thus, begin by finding the general solution of,
$\displaystyle y'-y^{-1}=0$
Thus,
$\displaystyle y'=y^{-1}$
Note, this is a Bernoulli, equation
Use substitution,
$\displaystyle u=y^2$
Thus,
$\displaystyle \frac{du}{dx}=\frac{du}{dy}\cdot \frac{dy}{dx}$.
Thus,
$\displaystyle \frac{du}{dx}=2y(y^{-1})=2$
Thus,
$\displaystyle u=2x+C$
Thus,
$\displaystyle y^2=2x+C$
Thus,
$\displaystyle y=\sqrt{2x+C}$---> general solution.

For the particiluar solution for this ode I had trouble. I went threw many different attempt non of them work. So maybe this is no elementary function which satisfies this particular condition.

7. non-homogeneous [...] particiluar solution

I don't think that method applies here, because the equation is not linear.

8. Originally Posted by Rebesques
I don't think that method applies here, because the equation is not linear.
That bothered me too, I chose to ignore it.

9. That bothered me too, I chose to ignore it.

...U are the man!!!

10. Originally Posted by ThePerfectHacker
That bothered me too, I chose to ignore it.
We'll make a Physicist out of this guy yet!

-Dan

11. I only do that when it comes to diffrencial equations.