Ad
related to: non homogeneous differential equation definition
Search results
Results from the WOW.Com Content Network
Consider a linear non-homogeneous ordinary differential equation of the form = + (+) = where () denotes the i-th derivative of , and denotes a function of .. The method of undetermined coefficients provides a straightforward method of obtaining the solution to this ODE when two criteria are met: [2]
The order of the differential equation is the highest order of derivative of the unknown function that appears in the differential equation. For example, an equation containing only first-order derivatives is a first-order differential equation, an equation containing the second-order derivative is a second-order differential equation, and so on.
Continuous group theory, Lie algebras, and differential geometry are used to understand the structure of linear and non-linear (partial) differential equations for generating integrable equations, to find its Lax pairs, recursion operators, Bäcklund transform, and finally finding exact analytic solutions to DE.
A linear differential equation that fails this condition is called inhomogeneous. A linear differential equation can be represented as a linear operator acting on y(x) where x is usually the independent variable and y is the dependent variable. Therefore, the general form of a linear homogeneous differential equation is =
Because the Wronskian is non-zero, the two functions are linearly independent, so this is in fact the general solution for the homogeneous differential equation (and not a mere subset of it). We seek functions A(x) and B(x) so A(x)u 1 + B(x)u 2 is a particular solution of the non-homogeneous equation. We need only calculate the integrals
The ERF method of finding a particular solution of a non-homogeneous differential equation is applicable if the non-homogeneous equation is or could be transformed to form () = + + +; where , are real or complex numbers and () is homogeneous linear differential equation of any order. Then, the exponential response formula can be applied to each ...
Therefore if the homogeneous equation has nontrivial solutions, multiple Green's functions exist. In some cases, it is possible to find one Green's function that is nonvanishing only for s ≤ x {\displaystyle s\leq x} , which is called a retarded Green's function, and another Green's function that is nonvanishing only for s ≥ x ...
If a non-homogeneous linear difference equation has been converted to homogeneous form which has been analyzed as above, then the stability and cyclicality properties of the original non-homogeneous equation will be the same as those of the derived homogeneous form, with convergence in the stable case being to the steady-state value y* instead ...
Ad
related to: non homogeneous differential equation definition