The origin is the only critical point of the nonlinear second-order differential equation x" + (x)² + x = 0. (a) Show that the phase-plane method leads to the Bernoulli differential equation. dy =-y-xy-¹. dx Let dx dt = y. Then the differential equation is dy = dx 1 (b) Show that the solution satisfying x(0) = and x'(0) 0 is not periodic. Let w = y². Using the chain rule and the differential equation from part (a), it follows that dw + 2w = -2x, which is a linear first order differential equation whose solution is y² = w = dx Since x(0) = and y(0) = x'(0) = 0, the particular solution is y² = , a parabola with vertex at (x, y) = Therefore the solution X(t) with X(0) = not periodic. is
The origin is the only critical point of the nonlinear second-order differential equation x" + (x)² + x = 0. (a) Show that the phase-plane method leads to the Bernoulli differential equation. dy =-y-xy-¹. dx Let dx dt = y. Then the differential equation is dy = dx 1 (b) Show that the solution satisfying x(0) = and x'(0) 0 is not periodic. Let w = y². Using the chain rule and the differential equation from part (a), it follows that dw + 2w = -2x, which is a linear first order differential equation whose solution is y² = w = dx Since x(0) = and y(0) = x'(0) = 0, the particular solution is y² = , a parabola with vertex at (x, y) = Therefore the solution X(t) with X(0) = not periodic. is
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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Transcribed Image Text:The origin is the only critical point of the nonlinear second-order differential equation x" + (x')² + x = 0.
(a) Show that the phase-plane method leads to the Bernoulli differential equation
dy
dx
= -y - xy-¹.
dx
Let = y. Then the differential equation is
dt
dy
dx
(b) Show that the solution satisfying x(0) =
and x'(0) = 0 is not periodic.
Let w = y². Using the chain rule and the differential equation from part (a), it follows that dw + 2w = -2x, which is a linear first order differential equation whose solution is y²
= W =
dx
Since x(0) = 1/2 and y(0) = x'(0) = 0, the particular solution is y²
=
, a parabola with vertex at (x, y) =
Therefore the solution X(t) with X(0) =
not periodic.
is
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