consider the nonhomogeneous linear system { dr * = a1 (t)x + b1(t)y + fi(t). * = a2(t)x + b2(t)y + f2(t).
consider the nonhomogeneous linear system { dr * = a1 (t)x + b1(t)y + fi(t). * = a2(t)x + b2(t)y + f2(t).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:consider the nonhomogeneous linear system
{
dr
# = a1 (t)x + bị1(t)y + fi(t).
# = a2 (t)x + b2(t)y + f2(t).
(*)
and the corresponding homogeneous system
{
dr
* = a1 (t)x + b1(t)y.
fip
* = a2(t)x + b2(t)y.
||
(**)
a) Suppose that
Sr = x1(t)
ly = y1(t).
x = r2(t)
ly = y2(t).
and
are linearly independent solutions of
equation(**).so, that
x = c1x1(t) + c2r2(t)
y = c1y1(t) + c2y2(t).
||
is its general solution. Then show that
x = vi(t)x1(t) + v2r2(t)
y = v1(t)y1(t) + v2(t)y2(t).
is a particular solution of Equation(*) if the
function v1,v2 satisfy the system
víy1+v2¥2 (t)=f2.
this technique for finding
particular solutions of nonhomogeneous
linear system is called the method of variation
of parameters.
b) Apply the method described in part (a) to
find particular solution of the
nonhomogeneous system
IP
= x + y – 5t +2.
dt
3D 4х —
4.x
2у — 8t — 8.
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