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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.
In mathematics, Liouville's formula, also known as the Abel–Jacobi–Liouville identity, is an equation that expresses the determinant of a square-matrix solution of a first-order system of homogeneous linear differential equations in terms of the sum of the diagonal coefficients of the system.
For an arbitrary system of ODEs, a set of solutions (), …, are said to be linearly-independent if: + … + = is satisfied only for = … = =.A second-order differential equation ¨ = (,, ˙) may be converted into a system of first order linear differential equations by defining = ˙, which gives us the first-order system:
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the ...
Although this is defined using a particular coordinate system, the transformation law relating the ξ i and the x i ensures that σ P is a well-defined function on the cotangent bundle. The function σ P is homogeneous of degree k in the ξ variable. The zeros of σ P, away from the zero section of T ∗ X, are the characteristics of P.
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 ...
In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems .
At i = 1 and n − 1 there is a term involving the boundary values () = and () = and since these two values are known, one can simply substitute them into this equation and as a result have a non-homogeneous system of linear equations that has non-trivial solutions.