-2.2.35 Show that dy/dx=(x+3y)/(x-y) is homogeneous. and....

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In summary, a homogeneous differential equation has terms with the same degree and follows the form F(x,y) = 0. To determine if an equation is homogeneous, one can check the degree of each term or use the substitution method to rewrite it with terms of the same degree. To show that an equation is homogeneous, the substitution method can be used to simplify it to have only terms of the same degree. The given equation dy/dx=(x+3y)/(x-y) demonstrates homogeneity as it can be rewritten in the form F(x,y) = 0 and can be simplified using the substitution method. Working with homogeneous differential equations has benefits such as being easier to solve, having a wider range of techniques, and having physical significance
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karush
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$\dfrac{dy}{dx}=\dfrac{x+3y}{x-y}$
ok well following the book example: divide numerator and denominator by x

$\dfrac{dy}{dx}=\dfrac{1+3\dfrac{y}{x}}{1-\dfrac{y}{x}}$

apparently, thus this is homogeneous but not sure why?

next solve the DE:unsure:
 
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1. What is the definition of a homogeneous function?

A homogeneous function is a mathematical function that satisfies the property of homogeneity, which means that if all of its variables are multiplied by a constant, the function's value will also be multiplied by that same constant.

2. How do you prove that a function is homogeneous?

To prove that a function is homogeneous, you must show that it satisfies the definition of homogeneity. In other words, you must demonstrate that when all of the function's variables are multiplied by a constant, the function's value is also multiplied by that same constant.

3. How do you show that the given function is homogeneous?

To show that the given function, dy/dx=(x+3y)/(x-y), is homogeneous, you must demonstrate that when all of its variables (x and y) are multiplied by a constant, the function's value is also multiplied by that same constant. This can be done by substituting cx for x and cy for y in the function and simplifying to show that the result is equal to c times the original function.

4. What is the importance of proving that a function is homogeneous?

Proving that a function is homogeneous is important because it allows us to use certain techniques and methods, such as Euler's homogeneous function theorem, to solve differential equations involving that function. It also helps us better understand the behavior and properties of the function.

5. Are there any other ways to prove that a function is homogeneous?

Yes, there are other ways to prove that a function is homogeneous. One method is to use the definition of homogeneity and show that the function satisfies it. Another method is to use the properties of homogeneous functions, such as Euler's homogeneous function theorem, to show that the function is homogeneous.

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