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Lecture 6 Solving of Heterogeneous Linear Differential
Equations (LNDE) With Constant Coefficients

Let us take linear heterogeneous differential equation of
the n-th order (LNDE):

y (n )  a y (n ) 1  ... a y   a y  f (x ) (6.1)
1 n 1 n

As known, solving of linear heterogeneous equation is
erected:
1) before finding of the fundamental system of decisions
of the proper homogeneous equation ;
2) before finding of even one partial decision of the set
heterogeneous equation.
The first from these tasks is considered on a previous
lecture. We will consider the methods of finding of partial
decisions of LNDE.

6.1 Method of variation

The method of variation of arbitrary constant is offered
by Lagrange and used for finding of general decisions of LNDE,
if general decision of LHDE is known. We will consider in
detail this method on the example of equations of the 2-nd order.

 y  a y   a y  0 (6.2)
1 2

Lets we know his common decision LHDE.


y  C y  C y , (6.3)
1 1 2 2

where y , y are one of the fundamental systems of decisions
1 2
of equation C and C are arbitrary constant.
1 2
We will put such task : to find the decision part LNDE – 2

y   a y   a y  f (x ) (6.4)
1 2
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