who to identify that the given differential equation is homogeneous and linear
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I think it was homogenous if it didnt have a constant in the equation. And its linear if it doesnt have any of the \(y^2, y^3\) etc. or \(y'*y\) or \(y'^2\). I'm not sure if \(x^2\) parts ruins the linearity of the equation.
can you tell me what homogeneous equation actually means
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Linear: the equation contains only y' or y terms, without y^2, y^4, (y')^2 etc in the equation. Homogeneous
Examples: \(y′+4x^2y=0\) is a homogeneous differential equation.
\(y′+6x^2y=5 \) is inhomogeneous
For a linear equation, if you have non-zero terms that do not contain y or y', it is not homogeneous.
the second equation in the example you gave is linear ?
there are 2 definitions of homogenous
the usual definition meaning: this = 0
the other is: f(x,y) such that T(tx,ty) = t f(x,y) or some such