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If u is a vector representing a solution to a homogeneous system, and r is any scalar, then ru is also a solution to the system. These are exactly the properties required for the solution set to be a linear subspace of R n. In particular, the solution set to a homogeneous system is the same as the null space of the corresponding matrix A.
A linear differential equation is homogeneous if it is a homogeneous linear equation in the unknown function and its derivatives. It follows that, if φ(x) is a solution, so is cφ(x), for any (non-zero) constant c. In order for this condition to hold, each nonzero term of the linear differential equation must depend on the unknown function or ...
The homogeneous (with all constant terms equal to zero) underdetermined linear system always has non-trivial solutions (in addition to the trivial solution where all the unknowns are zero). There are an infinity of such solutions, which form a vector space , whose dimension is the difference between the number of unknowns and the rank of the ...
A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is also true for a linear equation of order one, with non-constant coefficients.
In mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations ˙ = () is a matrix-valued function () whose columns are linearly independent solutions of the system. [1] Then every solution to the system can be written as () = (), for some constant vector (written as a column vector of height n).
In a finite-dimensional space, a homogeneous system of linear equations can be written as a single matrix equation: =. The set of solutions to this equation is known as the null space of the matrix. For example, the subspace described above is the null space of the matrix
We have given a homogeneous linear differential equation = of order with coefficients that are expandable as Laurent series with finite principle part. The goal is to obtain a fundamental set of formal Frobenius series solutions relative to any point ξ ∈ C {\displaystyle \xi \in \mathbb {C} } .
Homogeneous system: Homogeneous system of linear algebraic equations; System of homogeneous differential equations. System of homogeneous first-order differential equations; System of homogeneous linear differential equations; Homogeneous system in physics
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