In this problem we indicate an alternative procedure for solving the differential equation y" + by' + cy = (D² + bD+ c) y = g(t), (4) %3D 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 r + r2 = -b and rir2 = c. = (D – r2) y. Then show that the solution of Eq.(1) can be solved by solving the (b) Let u following two first order equations: (D – ri) 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) %3D 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 r + r2 = -b and rir2 = c. = (D – r2) y. Then show that the solution of Eq.(1) can be solved by solving the (b) Let u following two first order equations: (D – ri) 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
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I only need help on part (b). Thanks.
![**Solving Second-Order Linear Differential Equations**
In this problem, we explore an alternative procedure for solving the differential equation:
\[ y'' + by' + cy = (D^2 + bD + c)y = g(t), \tag{4} \]
where \( b \) and \( c \) are constants, and \( D \) denotes differentiation with respect to \( t \). Let \( r_1 \) and \( r_2 \) 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) Verification of the Factored Form
Verify that Eq. (1) can be written in the factored form:
\[ (D - r_1)(D - r_2)y = g(t), \]
where \( r_1 + r_2 = -b \) and \( r_1r_2 = c \).
### (b) Solving the Factored Equation
Let \( u = (D - r_2)y \). Then show that the solution of Eq. (1) can be solved by solving the following two first-order equations:
\[ (D - r_1)u = g(t), \]
\[ (D - r_2)y = u(t). \]
By converting the second-order differential equation into a system of first-order differential equations, we simplify the process of finding the solution \( y \). This method involves solving for \( u \) first and then \( y \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88e1e2e4-888b-4182-8c02-fd46dda7f6b1%2F2ad631e3-bd19-4fa2-9f1b-e19507687893%2Fswel1ra_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Solving Second-Order Linear Differential Equations**
In this problem, we explore an alternative procedure for solving the differential equation:
\[ y'' + by' + cy = (D^2 + bD + c)y = g(t), \tag{4} \]
where \( b \) and \( c \) are constants, and \( D \) denotes differentiation with respect to \( t \). Let \( r_1 \) and \( r_2 \) 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) Verification of the Factored Form
Verify that Eq. (1) can be written in the factored form:
\[ (D - r_1)(D - r_2)y = g(t), \]
where \( r_1 + r_2 = -b \) and \( r_1r_2 = c \).
### (b) Solving the Factored Equation
Let \( u = (D - r_2)y \). Then show that the solution of Eq. (1) can be solved by solving the following two first-order equations:
\[ (D - r_1)u = g(t), \]
\[ (D - r_2)y = u(t). \]
By converting the second-order differential equation into a system of first-order differential equations, we simplify the process of finding the solution \( y \). This method involves solving for \( u \) first and then \( y \).
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