31.Exact Equations. The equation 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 as in Section 2.1. By equating the coefficients of the preceding equations and then eliminating f(x), show that a necessary condition for exactness is P′′(x)−Q′(x)+R(x)=0 It can be shown that this is also a sufficient condition.
31.Exact Equations. The equation 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 as in Section 2.1. By equating the coefficients of the preceding equations and then eliminating f(x), show that a necessary condition for exactness is P′′(x)−Q′(x)+R(x)=0 It can be shown that this is also a sufficient condition.
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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31.Exact Equations. The equation
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 as in Section 2.1. By equating the coefficients of the preceding equations and then eliminating f(x), show that a necessary condition for exactness is
P′′(x)−Q′(x)+R(x)=0
It can be shown that this is also a sufficient condition.
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