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4. Prove that if o is a solution of the integral equation M1) = e" +a sin (t – s) y(s) ds (assuming the existence of the integral), then o satisfies the differential equation y" + (1 + a/t?)y = 0.

4. Prove that if o is a solution of the integral equation M1) = e" +a sin (t – s) y(s) ds (assuming the existence of the integral), then o satisfies the differential equation y" + (1 + a/t?)y = 0.

Advanced Engineering Mathematics
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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this question is from Qualitative Theory Of  Ordinary Differential Equations - Brauer and Nohel, exercise 13 section 3.1

13. Consider the integral equation y(s) (1) = e" +a[ sin (t -s) ds of Exercise 4. Define the successive approximations (do(t) = 0 - eit. Pn -1(s) ds (1<I< 0) sin (t s) (a) Show by induction that (n – 1)! t"-1 (1<1< 0;n = 1, 2, ...) Since ,(1) = do(t) + (41 – þo) + ·…· + (4,(t) – þ. - 1(1)) this shows that the , are well defined for 1<t< ∞, and {d,} converge uniformly for 1<i<o to a continuous limit function 6. (b) Show that the limit function satisfies the given integral equation. 3.1 Existence in the Scalar Case 119 (c) Using and the above estimate for 4.(t) - $a-1(t)), show that the limit function satisfies the estimate
Transcribed Image Text:13. Consider the integral equation y(s) (1) = e" +a[ sin (t -s) ds of Exercise 4. Define the successive approximations (do(t) = 0 - eit. Pn -1(s) ds (1<I< 0) sin (t s) (a) Show by induction that (n – 1)! t"-1 (1<1< 0;n = 1, 2, ...) Since ,(1) = do(t) + (41 – þo) + ·…· + (4,(t) – þ. - 1(1)) this shows that the , are well defined for 1<t< ∞, and {d,} converge uniformly for 1<i<o to a continuous limit function 6. (b) Show that the limit function satisfies the given integral equation. 3.1 Existence in the Scalar Case 119 (c) Using and the above estimate for 4.(t) - $a-1(t)), show that the limit function satisfies the estimate
4. Prove that if o is a solution of the integral equation t) = e" +a sin (1 - 3) y(s) ds (assuming the existence of the integral), then o satisfies the differential equation y" + (1 + a/t?)y = 0.
Transcribed Image Text:4. Prove that if o is a solution of the integral equation t) = e" +a sin (1 - 3) y(s) ds (assuming the existence of the integral), then o satisfies the differential equation y" + (1 + a/t?)y = 0.
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