Determine if a First-Order Differential Equation is Homogeneous - Part 2

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This video explains how to determine if a given linear first order differential equation is homogeneous.

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You treated this concept very well by explaining how to test for homogeneity of a function and next, by extension homogeneity of a Differential Equation (DE) by showing ways to test for homogeneity by definitions presented in two different forms or two different tests, with the definition expressed in two different forms. Excellent! But in practical terms, what does homogeneous equations of any sort mean and what is the importance of it conceptionally and how can we use it or what is the take-away benefit in math in general and in solving problems specifically besides using it to help separate variables to produce a simpler DE to find a solution to the Differential equation?

toniojz

Dear Mathispower4u,
Thank you for these video.
We've learnt in another video how to determine whether a function is homogeneous of degree n. E.g. y^4 / x^2 is homogeneous of degree 2 because f(tx, ty) = t^2 x f(x, y). However, we cannot write y^4 / x^2 as a function of y/x, so the DE dy/dz = y^4 / x^2 is not a homogeneous DE. Where is the catch? Is it that when we talk about a homogeneous DE, we mean homogeneous DE of the first degree? Thank you.

c.riccio