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

In summary, somebody solved this differential equation by looking at it in a "nicer" form and inputting the separation of variables. They also posted a picture of the equation in LaTeX form.
  • #1
drdolittle
27
0
somebody slove this differential equations

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

thanx in advance
 
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  • #2
Perhaps looking at it like this:

[tex]\frac{1}{\frac{dy}{dx}} = \frac{1}{y} + \frac{1}{x}[/tex]

[tex]\frac{dx}{dy} = \frac{1}{y} + \frac{1}{x}[/tex]

[tex]x\frac{dx}{dy} = \frac{x}{y} + 1[/tex]

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)
 
  • #3
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
 
  • #4
Well can you post what you have done please so others can help.
 
  • #5
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.
 
Last edited:
  • #6
try separation of variables...after that iam struggling to cotinue...
 
  • #7
Even though I just started learning differential equations, I thought I'd give this a try:

[tex]\frac{dx}{dy}=\frac{1}{y}+\frac{1}{x}[/tex]

[tex]\frac{dy}{dx}=\frac{xy}{x+y}[/tex]

[tex]x\frac{dy}{dx}+y\frac{dy}{dx}=xy[/tex]

[tex]y+x\frac{dy}{dx}+y\frac{dy}{dx}=y+xy[/tex]

[tex]\frac{d(xy)}{dx}+\frac{1}{2}\frac{d(y^2)}{dx}=y(1+x)[/tex]

[tex]\frac{1}{y}\,d(xy)+\frac{1}{2y}\,d(y^2)=(1+x)\,dx[/tex]

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.
 
  • #8
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:

[tex]\frac{d(xy)}{dx} = x\frac{dy}{dx} + y[/tex]
 
  • #9
I added a y to the LHS in the 4th step.
 
  • #10
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.
 
  • #11
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 :wink:
 
  • #12
iSamer said:
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 :wink:

They use LaTeX. See this thread for more info :smile:.
 

1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves the use of derivatives to represent the rate of change of a variable in a given system.

2. What is the purpose of differential equations?

The purpose of differential equations is to model and analyze real-world problems in various fields such as physics, engineering, economics, and biology. They provide a powerful tool for predicting the behavior of complex systems.

3. What are the types of differential equations?

The three main types of differential equations are ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). ODEs involve a single independent variable, PDEs involve multiple independent variables, and SDEs involve random processes.

4. How are differential equations solved?

Differential equations can be solved analytically or numerically. Analytical solutions involve finding an explicit formula for the function that satisfies the equation, while numerical solutions involve using a computer to approximate the solution.

5. What are some real-world applications of differential equations?

Differential equations have a wide range of applications, including modeling population growth, predicting weather patterns, designing electrical circuits, and analyzing financial markets. They are also used in fields such as medical imaging, control theory, and quantum mechanics.

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