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Given a homogeneous polynomial of degree with real coefficients that takes only positive values, one gets a positively homogeneous function of degree / by raising it to the power /. So for example, the following function is positively homogeneous of degree 1 but not homogeneous: ( x 2 + y 2 + z 2 ) 1 2 . {\displaystyle \left(x^{2}+y^{2}+z^{2 ...
A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written (,) = (,), where f and g are homogeneous functions of the same degree of x and y. [1] In this case, the change of variable y = ux leads to an equation of the form
This suggests that certain values of r will allow multiples of e rx to sum to zero, thus solving the homogeneous differential equation. [5] In order to solve for r, one can substitute y = e rx and its derivatives into the differential equation to get
Just as with any linear system of two equations, two solutions may be called linearly-independent if + = implies = =, or equivalently that | ˙ ˙ | is non-zero. This notion is extended to second-order systems, and any two solutions to a second-order ODE are called linearly-independent if they are linearly-independent in this sense.
Every homogeneous system has at least one solution, known as the zero (or trivial) solution, which is obtained by assigning the value of zero to each of the variables. If the system has a non-singular matrix (det(A) ≠ 0) then it is also the only solution. If the system has a singular matrix then there is a solution set with an infinite number ...
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. [1] For example, x 5 + 2 x 3 y 2 + 9 x y 4 {\displaystyle x^{5}+2x^{3}y^{2}+9xy^{4}} is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5.
For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration, [24] meaning here that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after. These finite-duration solutions can't be analytical functions on the whole real line, and because ...
The multi-homogeneous Bézout bound on the number of solutions may be used for non-homogeneous systems of equations, when the polynomials may be (multi)-homogenized without increasing the total degree. However, in this case, the bound may be not sharp, if there are solutions "at infinity".