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e the phase-plane method to show that the solution to the nonlinear second-order differential equation x" + 2x - x² = 0 that satisfies x(0) = 1 and x'(0) = 0 is periodic. 1,2 dy dx dt = y. Then the differential equation can be solved by separating variables. It follows that the general solution is 2/₁² = C+ x³ = x² dx y , and since X 0) = (x(0), x'(0)) = (1, 0) then the particular solution is But for each x such that 1-√√3 < x < 1, the particular solution has 1 X lue(s) of y. Therefore X(t) is a periodic solution. = 2/² = (x − 1) (x² – 2x − 2) X correspondi

e the phase-plane method to show that the solution to the nonlinear second-order differential equation x" + 2x - x² = 0 that satisfies x(0) = 1 and x'(0) = 0 is periodic. 1,2 dy dx dt = y. Then the differential equation can be solved by separating variables. It follows that the general solution is 2/₁² = C+ x³ = x² dx y , and since X 0) = (x(0), x'(0)) = (1, 0) then the particular solution is But for each x such that 1-√√3 < x < 1, the particular solution has 1 X lue(s) of y. Therefore X(t) is a periodic solution. = 2/² = (x − 1) (x² – 2x − 2) X correspondi

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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Use the phase-plane method to show that the solution to the nonlinear second-order differential equation x" + 2x - x² = 0 that satisfies x(0) = 1 and x'(0) = 0 is periodic. x² - 2x dx Let = y. Then the differential equation dt dy dx can be solved by separating variables. It follows that the general solution is 2=C+ x³ = x² and since y X(0) = (x(0), x'(0)) = (1, 0) then the particular solution is . But for each x such that 1 -√3 < x < 1, the particular solution has 1 value(s) of y. Therefore X(t) is a periodic solution. = 27=(x = (x − 1) (x² – 2x − 2) X corresponding
Transcribed Image Text:Use the phase-plane method to show that the solution to the nonlinear second-order differential equation x" + 2x - x² = 0 that satisfies x(0) = 1 and x'(0) = 0 is periodic. x² - 2x dx Let = y. Then the differential equation dt dy dx can be solved by separating variables. It follows that the general solution is 2=C+ x³ = x² and since y X(0) = (x(0), x'(0)) = (1, 0) then the particular solution is . But for each x such that 1 -√3 < x < 1, the particular solution has 1 value(s) of y. Therefore X(t) is a periodic solution. = 27=(x = (x − 1) (x² – 2x − 2) X corresponding
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