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P(x)y" + Q(x)y' + R(x)y = 0 is said to be exact if it can be written in the form (P(x)y')' + (f(x)y)' = 0, where f(x) is to be determined in terms of P(x), Q(x), and R(x). The latter equation can be integrated once immediately, resulting in a first-order linear equation for y that can be solved easily. (a) By equating the coefficients of the preceding equations and then eliminating f(x), show that the equation is exact if P" (x) - Q'(x) + R(x) = 0. (b). By above observations solve the equation xy" — (cos x)y' + (sin x)y = 0, x > 0

P(x)y" + Q(x)y' + R(x)y = 0 is said to be exact if it can be written in the form (P(x)y')' + (f(x)y)' = 0, where f(x) is to be determined in terms of P(x), Q(x), and R(x). The latter equation can be integrated once immediately, resulting in a first-order linear equation for y that can be solved easily. (a) By equating the coefficients of the preceding equations and then eliminating f(x), show that the equation is exact if P" (x) - Q'(x) + R(x) = 0. (b). By above observations solve the equation xy" — (cos x)y' + (sin x)y = 0, x > 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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P(x)y" + Q(x)y' + R(x)y = 0 is said to be exact if it can be written in the form (P(x)y')' + (f(x)y)' = 0, where f(x) is to be determined in terms of P(x), Q(x), and R(x). The latter equation can be integrated once immediately, resulting in a first-order linear equation for y that can be solved easily. (a) By equating the coefficients of the preceding equations and then eliminating f(x), show that the equation is exact if P" (x) - Q'(x) + R(x) = 0. (b). By above observations solve the equation xy" — (cos x)y' + (sin x)y = 0, x > 0
Transcribed Image Text:P(x)y" + Q(x)y' + R(x)y = 0 is said to be exact if it can be written in the form (P(x)y')' + (f(x)y)' = 0, where f(x) is to be determined in terms of P(x), Q(x), and R(x). The latter equation can be integrated once immediately, resulting in a first-order linear equation for y that can be solved easily. (a) By equating the coefficients of the preceding equations and then eliminating f(x), show that the equation is exact if P" (x) - Q'(x) + R(x) = 0. (b). By above observations solve the equation xy" — (cos x)y' + (sin x)y = 0, x > 0
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