In this problem we indicate an alternative procedure for solving the differential equation y" + by' + cy = (D² + bD + c) y = g(t), (4) where b and c are constants, and D denotes differentiation with respect to t. Let ri and r2 be the zeros of the characteristic polynomial of the corresponding homogeneous equation. These roots may be real and different, real and equal, or conjugate complex numbers. (a) Verify that Eq. (1) can be written in the factored form (D – ri)(D – r2) y = g(t), where ri + r2 = -b and rır2 = C. (b) Let u = (D – r2) y. Then show that the solution of Eq.(1) can be solved by solving the following two first order equations: (D – r1) u = g(t), (D– r2) y= u(t).
In this problem we indicate an alternative procedure for solving the differential equation y" + by' + cy = (D² + bD + c) y = g(t), (4) where b and c are constants, and D denotes differentiation with respect to t. Let ri and r2 be the zeros of the characteristic polynomial of the corresponding homogeneous equation. These roots may be real and different, real and equal, or conjugate complex numbers. (a) Verify that Eq. (1) can be written in the factored form (D – ri)(D – r2) y = g(t), where ri + r2 = -b and rır2 = C. (b) Let u = (D – r2) y. Then show that the solution of Eq.(1) can be solved by solving the following two first order equations: (D – r1) u = g(t), (D– r2) y= u(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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I only need help on part (b)
Transcribed Image Text:In this problem we indicate an alternative procedure for solving the differential equation
y" + by' + cy = (D² +bD+ c) y = g(t),
(4)
where b and c are constants, and D denotes differentiation with respect to t. Let
be the zeros of the characteristic polynomial of the corresponding homogeneous equation.
These roots may be real and different, real and equal, or conjugate complex numbers.
and
r2
(a) Verify that Eq. (1) can be written in the factored form
(D – ri)(D – r2) y = g(t),
where ri + r2 = -b and rịr2= c.
(b) Let u = (D – r2) y. Then show that the solution of Eq.(1) can be solved by solving the
following two first order equations:
(D – ri) u = g(t),
(D – r2) y = u(t).
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