# Differential Equation (Homogeneous / scale-invariant

• PhysicsMark
In summary, the conversation is discussing how to test the equation (x+y^2)dy+ydx=0 for scale invariance and finding its general solutions. The speaker makes substitutions and solves for n, finding that n=1/2. They then make another substitution and attempt to find dy, but are unsure about the notation and ask for clarification. They eventually realize the correct form of dy is dy = v/(2√x)dx+√x dv.
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## Homework Statement

Test the following equation to show that they are scale invariant. Find their general solutions (It is not necessary to do the anti-derivative.)

$$(x+y^2)dy+ydx=0$$

I believe what my tutorial wants me to do is to check for homogeneity. I'm not sure though. This is not a Differential equations class, it is a math methods in physics class. The tutorial we use titles this section "Scale-invariant first-order differential equations".

## The Attempt at a Solution

First I make the following substitutions:

$$x=\alpha{x}$$

$$y=\alpha^{n}x$$

I then use the substitutions in the DiffEq and solve for n so that the weight of each term is equal. I find that:

$$n=\frac{1}{2}$$

Based off that, I now make the following substitution:

$$y=v\sqrt{x}$$

From there things begin to get messy. I am unsure about 1 step so far: I am not sure what dy equates to. In the case that n=1, dy = vdx+xdv. I am not sure how to get dy using n=1/2.

Is it:

$$dy= vdx+{\sqrt{x}}dv$$

or:

$$dy= \frac{v}{2\sqrt{x}}+{\sqrt{x}dv$$

Or something else completely?

I guess I am asking what is dx in the equations above? Is it the derivative of x^1/2 or is it just dx?

So it was something else completely. It is:

$$dy = \frac{v}{2\sqrt{x}}dx+{\sqrt{x}dv$$

I think I misunderstood basic differentiation notation.

## 1. What is a homogeneous differential equation?

A homogeneous differential equation is one where all the terms in the equation have the same degree. This means that all the variables have the same exponent. In other words, all the terms are of the same scale.

## 2. What does it mean for a differential equation to be scale-invariant?

A scale-invariant differential equation is one where the solution remains the same even if the variables are scaled by a constant factor. In other words, the solution is not affected by changes in scale.

## 3. How do you solve a homogeneous differential equation?

To solve a homogeneous differential equation, you can use the method of separation of variables. This involves isolating the variables on one side of the equation and integrating both sides to find the solution.

## 4. What is the general solution to a scale-invariant differential equation?

The general solution to a scale-invariant differential equation is a solution that is valid for all possible values of the variables. It is usually expressed in terms of arbitrary constants.

## 5. What role do initial conditions play in solving a scale-invariant differential equation?

Initial conditions are necessary to find the specific solution to a scale-invariant differential equation. These conditions specify the values of the variables at a certain point, which can then be used to determine the values of the arbitrary constants in the general solution.

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