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Equilibrium solutions and phase lines

Find the equilibrium solutions for each of the differential equations in Exercise Group 13Z8-12. Draw the phase line for each equation and classify each equilibrium solution as a sink, a source, or a node.

8. y’ = 2y – 5

dz/dt

(x – 1)(2 + 2)

dx

10. dy/dx

(22 – 1)(c – 2)

sin(2y)

12. y’ = (22 + 1)(2 – 1)

15. y’ = 22 + 1

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For each of the following differential equations, find all equilibrium solutions and determine if they are sinks, sources, or neither. Also, sketch the phase line.(a) x’ = x^3 – 3x(b) x’ = x^4 – x^2(c) x’ = cos(x)(d) x’ = sin(x)(e) x’ = 1 – x^2

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Hand-up question: Find all equilibrium points of dy = y^2 – 1)(1 – 2 dx and sketch the phase line. State whether the equilibrium points are sources, sinks, or nodes.

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10) Sketch the phase line for the following differential equation Identify the equilibrium points as sinks, sources, O nodes: d-y +2y 5 15 dt

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