this question is from Qualitative Theory Of Ordinary Differential Equations - Brauer and Nohel, exercise 13 section 3.1 13. Consider the integral equationy(s)(1) = e" +a[ sin (t -s) dsof Exercise 4. Define the successive approximations(do(t) = 0- eit.Pn -1(s)ds(1 4. Prove that if o is a solution of the integral equationt) = e" +a sin (1 - 3)y(s)ds(assuming the existence of the integral), then o satisfies the differential equationy" + (1 + a/t?)y = 0.

Given that,y(t)=eit+α∫t∞sin(t-s)y(s)s2ds

Answered: 4. Prove that if o is a solution of the…

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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

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

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.

With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.

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