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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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Please walk me through this problem. Thank you!
![In this problem, you will solve the non-homogeneous differential equation
\[ y'' + 49y = \sec^2(7x) \]
on the interval \(-\pi/14 < x < \pi/14\).
(1) Let \( C_1 \) and \( C_2 \) be arbitrary constants. The general solution of the related homogeneous differential equation \( y'' + 49y = 0 \) is the function
\[ y_h(x) = C_1 y_1(x) + C_2 y_2(x) = C_1 \, \underline{\hspace{1cm}} + C_2 \, \underline{\hspace{1cm}}. \]
(2) The particular solution \( y_p(x) \) to the differential equation \( y'' + 49y = \sec^2(7x) \) is of the form
\[ y_p(x) = y_1(x) \, u_1(x) + y_2(x) \, u_2(x) \]
where \( u_1'(x) = \underline{\hspace{1.5cm}} \) and \( u_2'(x) = \underline{\hspace{1.5cm}} \).
(3) It follows that
\[ u_1(x) = \underline{\hspace{1.5cm}} \, \text{and} \, u_2(x) = \underline{\hspace{1.5cm}}; \]
thus
\[ y_p(x) = \underline{\hspace{5cm}}. \]
(4) Therefore, on the interval \((- \pi/14, \pi/14)\), the most general solution of the non-homogeneous differential equation \( y'' + 49y = \sec^2(7x) \) is
\[ y = C_1 \, \underline{\hspace{1cm}} + C_2 \, \underline{\hspace{1cm}} + \underline{\hspace{5cm}}. \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F752c80f7-6a34-4004-b4db-3d79d0db3666%2F6c5c7e11-70c1-4108-9948-7160e05ddc2c%2Fsq3ta2p_processed.png&w=3840&q=75)
Transcribed Image Text:In this problem, you will solve the non-homogeneous differential equation
\[ y'' + 49y = \sec^2(7x) \]
on the interval \(-\pi/14 < x < \pi/14\).
(1) Let \( C_1 \) and \( C_2 \) be arbitrary constants. The general solution of the related homogeneous differential equation \( y'' + 49y = 0 \) is the function
\[ y_h(x) = C_1 y_1(x) + C_2 y_2(x) = C_1 \, \underline{\hspace{1cm}} + C_2 \, \underline{\hspace{1cm}}. \]
(2) The particular solution \( y_p(x) \) to the differential equation \( y'' + 49y = \sec^2(7x) \) is of the form
\[ y_p(x) = y_1(x) \, u_1(x) + y_2(x) \, u_2(x) \]
where \( u_1'(x) = \underline{\hspace{1.5cm}} \) and \( u_2'(x) = \underline{\hspace{1.5cm}} \).
(3) It follows that
\[ u_1(x) = \underline{\hspace{1.5cm}} \, \text{and} \, u_2(x) = \underline{\hspace{1.5cm}}; \]
thus
\[ y_p(x) = \underline{\hspace{5cm}}. \]
(4) Therefore, on the interval \((- \pi/14, \pi/14)\), the most general solution of the non-homogeneous differential equation \( y'' + 49y = \sec^2(7x) \) is
\[ y = C_1 \, \underline{\hspace{1cm}} + C_2 \, \underline{\hspace{1cm}} + \underline{\hspace{5cm}}. \]
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