I am to use the steps in the attached method to solve the differential equation: y'+(1/t)y=3 cos 2t, where t>0.
I am to use the steps in the attached method to solve the differential equation: y'+(1/t)y=3 cos 2t, where t>0.
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 am to use the steps in the attached method to solve the
y'+(1/t)y=3 cos 2t, where t>0.
![### Variation of Parameters
Consider the following method of solving the general linear equation of first order:
\[ y' + p(t)y = g(t) \tag{1} \]
#### (a)
If \( g(t) = 0 \) for all \( t \), show that the solution is
\[ y = A \exp \left[ - \int p(t) dt \right], \]
where \( A \) is a constant.
#### (b)
If \( g(t) \) is not everywhere zero, assume that the solution of Eq. (1) is of the form
\[ y = \Lambda(t) \exp \left[ - \int p(t) dt \right], \]
where \( \Lambda \) is a function of \( t \).
By substituting for \( y \) in the given differential equation, show that \( \Lambda(t) \) must satisfy the condition
\[ \Lambda'(t) = g(t) \exp \left[ \int p(t) dt \right]. \tag{4} \]
#### (c)
Find \( \Lambda(t) \) from Eq. (4). Then substitute for \( \Lambda(t) \) in Eq. (3) and determine \( y \). Verify that the solution obtained in this manner agrees with that of Eq. (33) in the text. Eq. (33) in the text is
\[ y = \frac{1}{\mu(t)} \left[ \int_{t_0}^t \mu(s)g(s) ds + c \right]. \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88e1e2e4-888b-4182-8c02-fd46dda7f6b1%2F5210342d-90f5-42d5-8d6c-28f7ba22bbb8%2Fva1d48o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Variation of Parameters
Consider the following method of solving the general linear equation of first order:
\[ y' + p(t)y = g(t) \tag{1} \]
#### (a)
If \( g(t) = 0 \) for all \( t \), show that the solution is
\[ y = A \exp \left[ - \int p(t) dt \right], \]
where \( A \) is a constant.
#### (b)
If \( g(t) \) is not everywhere zero, assume that the solution of Eq. (1) is of the form
\[ y = \Lambda(t) \exp \left[ - \int p(t) dt \right], \]
where \( \Lambda \) is a function of \( t \).
By substituting for \( y \) in the given differential equation, show that \( \Lambda(t) \) must satisfy the condition
\[ \Lambda'(t) = g(t) \exp \left[ \int p(t) dt \right]. \tag{4} \]
#### (c)
Find \( \Lambda(t) \) from Eq. (4). Then substitute for \( \Lambda(t) \) in Eq. (3) and determine \( y \). Verify that the solution obtained in this manner agrees with that of Eq. (33) in the text. Eq. (33) in the text is
\[ y = \frac{1}{\mu(t)} \left[ \int_{t_0}^t \mu(s)g(s) ds + c \right]. \]
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