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To solve this particular ordinary differential equation system, at some point in the solution process, we shall need a set of two initial values (corresponding to the two state variables at the starting point). In this case, let us pick x(0) = y(0) = 1.
Now () is an matrix solution of ′ =. This fundamental matrix will provide the homogeneous solution, and if added to a particular solution will give the general solution to the inhomogeneous equation. Let = be the general solution. Now,
If a term in the above particular integral for y appears in the homogeneous solution, it is necessary to multiply by a sufficiently large power of x in order to make the solution independent. If the function of x is a sum of terms in the above table, the particular integral can be guessed using a sum of the corresponding terms for y. [1]
A system of linear equations with n variables and coefficients in a field K has a solution if and only if its coefficient matrix A and its augmented matrix [A|b] have the same rank. [1] If there are solutions, they form an affine subspace of of dimension n − rank(A). In particular: if n = rank(A), the solution is unique,
5.5 Matrix solution. 5.6 Other methods. ... In particular, the solution set to a homogeneous system is the same as the null space of the corresponding matrix A.
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]
Synonym for binary matrix or logical matrix. Alternant matrix: A matrix in which successive columns have a particular function applied to their entries. Alternating sign matrix: A square matrix with entries 0, 1 and −1 such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign. Anti-diagonal ...
A singular solution y s (x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. The set on which a solution is singular may be as small as a single point or as large as the ...