ODE Change of Variable: Solving Separable Equations with u = y/x

In summary, the conversation discusses how to rewrite the ODE y' = f(y/x) as a separable equation using the change of variable u = y/x. The attempt at solution involves using the chain rule to rewrite y' in terms of u, but the resulting equation seems to be incorrect. The conversation ends with a question about how to correctly put the original equation into separable form.
  • #1
dipole
555
151

Homework Statement



I have the ODE [itex] y' = f(\frac{y}{x}) [/itex], and I want to re-write this as a separable equation using the change of variable [itex] u = \frac{y}{x} [/itex]

The Attempt at a Solution



I use the chain rule to write [itex] y' = \frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}
= \frac{dy}{du}(-\frac{y}{x^2}) = -\frac{dy}{du}\frac{u^2}{y} = f(u) [/itex]

which is a separable equation. However this seems to be wrong somehow because when I try using it to solve equations of the above form, I'm getting the wrong answer. Any help where I went wrong?
 
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  • #2
dipole said:

Homework Statement



I have the ODE [itex] y' = f(\frac{y}{x}) [/itex], and I want to re-write this as a separable equation using the change of variable [itex] u = \frac{y}{x} [/itex]

The Attempt at a Solution



I use the chain rule to write [itex] y' = \frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}
= \frac{dy}{du}(-\frac{y}{x^2}) = -\frac{dy}{du}\frac{u^2}{y} = f(u) [/itex]

which is a separable equation. However this seems to be wrong somehow because when I try using it to solve equations of the above form, I'm getting the wrong answer. Any help where I went wrong?

If [itex]u=y/x[/itex] then [itex]y'=x\frac{du}{dx}+u[/itex]
 
  • #3
I don't see how I can use that to put the original equation [itex] \frac{dy}{dx} = f(\frac{y}{x}) [/itex] into separable form though. :\
 
  • #4
Won't you then have the equation:

[tex]xu'+u=f(u)[/tex]

Ain't that separable?
 

Related to ODE Change of Variable: Solving Separable Equations with u = y/x

1. What is an ODE change of variable?

An ODE (ordinary differential equation) change of variable is a technique used to simplify a differential equation by substituting a new variable for the original variable. This can make the equation easier to solve or provide more insight into its behavior.

2. Why is an ODE change of variable useful?

ODE change of variable can be useful for solving difficult or complex differential equations. It can also help to identify important features of the equation, such as equilibrium points or special solutions.

3. How do you perform an ODE change of variable?

The process of performing an ODE change of variable involves substituting a new variable, usually denoted as u, for the original variable in the differential equation. This new variable should be chosen in a way that simplifies the equation or reveals important information about its behavior.

4. What are some common examples of ODE change of variable?

Some common examples of ODE change of variable include substitution of a trigonometric function, such as sine or cosine, for a variable, or using a logarithmic or exponential function to transform the equation. These changes of variables are often used to solve specific types of differential equations.

5. Is there a formula for performing an ODE change of variable?

No, there is no one formula for performing an ODE change of variable. The choice of the new variable depends on the specific equation and the desired outcome. It often involves trial and error or knowledge of common substitutions for certain types of equations.

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