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 thatSr = x1(t)ly = y1(t).x = r2(t)ly = y2(t).andare linearly independent solutions ofequation(**).so, thatx = c1x1(t) + c2r2(t)y = c1y1(t) + c2y2(t).||is its general solution. Then show thatx = vi(t)x1(t) + v2r2(t)y = v1(t)y1(t) + v2(t)y2(t).is a particular solution of Equation(*) if thefunction v1,v2 satisfy the systemvíy1+v2¥2 (t)=f2.this technique for findingparticular solutions of nonhomogeneouslinear system is called the method of variationof parameters.b) Apply the method described in part (a) tofind particular solution of thenonhomogeneous systemIP= x + y – 5t +2.dt3D 4х —4.x2у — 8t — 8.

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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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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

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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