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The following is the textbook extract while it shows that if the right side of a equation can be expressed as function of ratio $(y/x)$, then the equation is homogeneous and a homogeneous equation can always be represented in a separable form.

... This means that integral curves have the same slope at all points on any given straight line through the origin, although the slope changes from one line to another. Therefore the direction field and the integral curves are symmetric with respect to the origin.

I did not understand the conclusion of the slope, is it so that all the homogeneous DE integral curves are symmetric about some straight line through origin?

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closed as not a real question by Did, t.b., Noah Snyder, Arkamis, no identity Oct 13 '12 at 17:46

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Your equation (1) doesn't match its description. –  Robert Israel Jul 2 '12 at 5:24
    
NARQ. $ $ $ $ $ $ –  Did Jul 2 '12 at 6:19

1 Answer 1

up vote 2 down vote accepted

The integral curves have the same slope on the straight lines passing through the origin.

Actually, it is more simple to start with the equations of the form $$ \frac{dx}{dt}=f(x), $$ which is called autonomous, because the right hand side does not depend on $t$ explicitly. What special about this equation? Answer: If $x(t)$ is a solution for the initial condition $x(0)=x_0$, then $x(t+C)$ is a solution for the initial condition $x(C)=x_0$. In different words, all the solutions can be obtained from one by translating to the left or right. And yet in another words: if you plot your solution curves of this equation, the slope on the straight lines parallel to the time axis is the same (because they are the same curves, obtained by translation!) I leave you make a picture yourself. Note that sometimes the straight lines $x(t)=\hat{x}$ are solutions, and in this case they separate families of other solutions.

For the homogeneous equations similar facts are valid if you replace straight lines parallel to the time axis by the straight lines passing through the origin. That is, the integral curves of the homogeneous equation are obtained with the expanding and shrinking of one solution. That is, the slope on the integral curves on the straight lines through the origin is the same (see the first paragraph of the Arnol's book Geometric methods...)

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