Formulas for computing the vector of a particular solution of inhomogeneous system of differential equations.

Instead of the formula for computing the vector of a particular solution of an inhomogeneous system of differential equations in the form [Gantmaher]:

it is proposed to use the following formula for each individual section of the integration interval:

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The correctness of the above formula is confirmed by the following:

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which was to be confirmed.

The calculation of the vector of a particular solution of a system of differential equations is performed using the representation of Cauchy’s matrix under the integral sign in the form of a series and integrating this series elementwise:

This formula is valid for the case of a system of differential equations with constant coefficient matrix =const.

Let us consider the variant, when the steps of the integration interval are chosen sufficiently small, which allows us to consider the vector  in the region  approximately as a constant , which allows us to remove this vector from the signs of the integrals:

It is known that when T=(at+b) we have

In our case, we have

Then we obtain .

Then we obtain a series for computing the vector of a particular solution of an inhomogeneous system of differential equations on a small section :

For the case of differential equations with variable coefficients, an averaged matrix  of the coefficients of the system of differential equations can be used for each section.

If the considered section of the integration interval is not small, then the following iterative (recurrent) formulas are proposed.

We give the formulas for computing the vector of a particular solution, for example,  on the considered section  through the vectors of the particular solution , , , corresponding to the subsections , , .

We have .

We also have a formula for a separate subsection:

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We can write:

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We substitute  in  and get:

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Let us compare the expression obtained with the formula:

and we get, obviously, that:

and for the particular vector we obtain the formula:

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That is, the subsector vectors  are not simply add with each other, but with the participation of Cauchy’s matrix of the subsection.

Similarly, we write down  and substitute the formula for  and get:          

Comparing the expression obtained with the formula:

obviously, we get that:

and together with this we get the formula for a particular vector:

That is, in this way a particular vector is calculated - the vector of the particular solution of the inhomogeneous system of differential equations, that is, for example, a particular vector  on the considered section  is calculated through the computed partial vectors , ,   corresponding to the sub-sections , , .

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