2. Consider the nonhomogeneous equation = a(t)y+b(t) and its associated homogeneous equation a(t)y. dt (a) Prove that if y(t) is a solution of (3) and yp(t) is a particular solution of (2), then y(t) + Up (t) is also a solution of (2). Prove that if yp, (t) and yp, (t) are two particular solutions of (2), then yp, (t)- yp. (t) is a solution of (3). dt
2. Consider the nonhomogeneous equation = a(t)y+b(t) and its associated homogeneous equation a(t)y. dt (a) Prove that if y(t) is a solution of (3) and yp(t) is a particular solution of (2), then y(t) + Up (t) is also a solution of (2). Prove that if yp, (t) and yp, (t) are two particular solutions of (2), then yp, (t)- yp. (t) is a solution of (3). dt
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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![### Topic: Nonhomogeneous Differential Equations
#### Problem 2
Consider the nonhomogeneous equation:
\[ \frac{dy}{dt} = a(t)y + b(t) \]
and its associated homogeneous equation:
\[ \frac{dy}{dt} = a(t)y \]
(a) **Problem Statement:**
Prove that if \( y_h(t) \) is a solution of the homogeneous equation (3) and \( y_p(t) \) is a particular solution of the nonhomogeneous equation (2), then \( y_h(t) + y_p(t) \) is also a solution of the nonhomogeneous equation (2).
(b) **Problem Statement:**
Prove that if \( y_{p_1}(t) \) and \( y_{p_2}(t) \) are two particular solutions of the nonhomogeneous equation (2), then \( y_{p_1}(t) - y_{p_2}(t) \) is a solution of the homogeneous equation (3).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9dbf69a1-1531-4733-a5dc-94a7f3821ec6%2F48552e53-c434-4c74-8ee3-9ca9495e619e%2Fi977uy8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Topic: Nonhomogeneous Differential Equations
#### Problem 2
Consider the nonhomogeneous equation:
\[ \frac{dy}{dt} = a(t)y + b(t) \]
and its associated homogeneous equation:
\[ \frac{dy}{dt} = a(t)y \]
(a) **Problem Statement:**
Prove that if \( y_h(t) \) is a solution of the homogeneous equation (3) and \( y_p(t) \) is a particular solution of the nonhomogeneous equation (2), then \( y_h(t) + y_p(t) \) is also a solution of the nonhomogeneous equation (2).
(b) **Problem Statement:**
Prove that if \( y_{p_1}(t) \) and \( y_{p_2}(t) \) are two particular solutions of the nonhomogeneous equation (2), then \( y_{p_1}(t) - y_{p_2}(t) \) is a solution of the homogeneous equation (3).
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