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Example 4. Find the equivalent Volterra integral equation to the following initial value problem Proceeding as before, we set y" (x) + y(x) = cos x, y(0) = 0, y (0) = 1. y" (x) = u(x). Integrating both sides of (102) from 0 to x, using the initial condition y'(0) = 1 yields fu(t)dt. Integrating (103), using the initial condition y(0) = 0 leads to or equivalently y (x): = 1+ y(x) = x + y(x) = x + u(x) the equivalent Volterra integral equation. S f* u(t) dtdt, √(x - (x-t)u(t) dt, (101) = cos xx- S. (2 (x-t)u(t) dt, 0 (102) upon using the conversion rule (71). Inserting (102) and (105) into (101) leads to the following required Volterra integral equation (103) (104) (105) (106)

Example 4. Find the equivalent Volterra integral equation to the following initial value problem Proceeding as before, we set y" (x) + y(x) = cos x, y(0) = 0, y (0) = 1. y" (x) = u(x). Integrating both sides of (102) from 0 to x, using the initial condition y'(0) = 1 yields fu(t)dt. Integrating (103), using the initial condition y(0) = 0 leads to or equivalently y (x): = 1+ y(x) = x + y(x) = x + u(x) the equivalent Volterra integral equation. S f* u(t) dtdt, √(x - (x-t)u(t) dt, (101) = cos xx- S. (2 (x-t)u(t) dt, 0 (102) upon using the conversion rule (71). Inserting (102) and (105) into (101) leads to the following required Volterra integral equation (103) (104) (105) (106)

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.1: Solutions Of Elementary And Separable Differential Equations
Problem 54E: Plant Growth Researchers have found that the probability P that a plant will grow to radius R can be...
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Example 4. Find the equivalent Volterra integral equation to the following initial value problem Proceeding as before, we set y" (x) + y(x) = cos x, y(0) = 0, y' (0) or equivalently -fu(t)dt. Integrating (103), using the initial condition y(0) = 0 leads to y" (x) = u(x). Integrating both sides of (102) from 0 to x, using the initial condition y'(0) = 1 yields y'(x): = 1+ y(x) = x + y(x) = = x + [ f* u(t) dtdt, 0 0 f (x − t)u(t) dt, = 1. (101) u(x) = cos x-x- - [(x - t)u(t) dt, 0 (102) (103) (104) upon using the conversion rule (71). Inserting (102) and (105) into (101) leads to the following required Volterra integral equation (105) (106) the equivalent Volterra integral equation. As previously remarked, linear Volterra integral equations will be discussed
Transcribed Image Text:Example 4. Find the equivalent Volterra integral equation to the following initial value problem Proceeding as before, we set y" (x) + y(x) = cos x, y(0) = 0, y' (0) or equivalently -fu(t)dt. Integrating (103), using the initial condition y(0) = 0 leads to y" (x) = u(x). Integrating both sides of (102) from 0 to x, using the initial condition y'(0) = 1 yields y'(x): = 1+ y(x) = x + y(x) = = x + [ f* u(t) dtdt, 0 0 f (x − t)u(t) dt, = 1. (101) u(x) = cos x-x- - [(x - t)u(t) dt, 0 (102) (103) (104) upon using the conversion rule (71). Inserting (102) and (105) into (101) leads to the following required Volterra integral equation (105) (106) the equivalent Volterra integral equation. As previously remarked, linear Volterra integral equations will be discussed
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