Solving Differential Equations to Solving 1/y' = (1/y)+(1/x)

[drdolittle]

somebody slove this differential equations

1/y' = (1/y)+(1/x)

thanx in advance

 

[Zurtex]

Perhaps looking at it like this:

$$\frac{1}{\frac{dy}{dx}} = \frac{1}{y} + \frac{1}{x}$$

$$\frac{dx}{dy} = \frac{1}{y} + \frac{1}{x}$$

$$x\frac{dx}{dy} = \frac{x}{y} + 1$$

lol, I'll stop there because I suddenly realize this is beyond me (but it looks in a 'nicer' form, perhaps it will help you)

 

[drdolittle]

Your solution is just a peanut compared to where i have gone...there is still more to go...anyhow thanks for trying,do try nmore and figure out the solution.

regards
drdolittle

 

[Zurtex]

Well can you post what you have done please so others can help.

 

[Zurtex]

I ran this through Mathematica: DSolve[1/(y'[x]) == 1/x + 1/y[x], y[x], x]

And it gave me nothing sorry.

Edit: Although I'm not used to using Mathematica and have yet to get it to solve the simplest thing I think I have inputed it right.

 

[drdolittle]

try separation of variables...after that iam struggling to cotinue...

 

[daster]

Even though I just started learning differential equations, I thought I'd give this a try:

$$\frac{dx}{dy}=\frac{1}{y}+\frac{1}{x}$$

$$\frac{dy}{dx}=\frac{xy}{x+y}$$

$$x\frac{dy}{dx}+y\frac{dy}{dx}=xy$$

$$y+x\frac{dy}{dx}+y\frac{dy}{dx}=y+xy$$

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

$$\frac{1}{y}\,d(xy)+\frac{1}{2y}\,d(y^2)=(1+x)\,dx$$

I don't know what to do now, and I don't know if any of this is right, but I hope it'd be of some use.

 

[Zurtex]

Err I still think this is beyond me but I think you made a mistake on the LHS going from the 4th to the 5th line as:

$$\frac{d(xy)}{dx} = x\frac{dy}{dx} + y$$

 

[daster]

I added a y to the LHS in the 4th step.

 

[Integral]

What I see when I look at that equation is a family of hyperbolas very much like the simple lens equation. There is a change of variables and a rotation that will reduce this equation to something which may be separable. Unfortunately I do not have the time to do all of the algebra for you.

Explore doing a change of variables, perhaps to polar coordinates, see what you get.

 

[iSamer]

How do you guys write the nice format of dy/dx and the fractions? Which program do you use, and you post them as photos?

I'll help in solving it, but after knowing how to post a math solution

 

[Nylex]

How do you guys write the nice format of dy/dx and the fractions? Which program do you use, and you post them as photos?

I'll help in solving it, but after knowing how to post a math solution

They use LaTeX. See this thread for more info .