18. Convert the second-order equation d²y 0 dt2 into a first-order system using v = dy/dt as usual. (a) Find the general solution for the dv/dt equation. (b) Substitute this solution into the dy/dt equation, and find the general solution of the system. (c) Sketch the phase portrait of the system.

18. Convert the second-order equation d²y 0 dt2 into a first-order system using v = dy/dt as usual. (a) Find the general solution for the dv/dt equation. (b) Substitute this solution into the dy/dt equation, and find the general solution of the system. (c) Sketch the phase portrait of the system.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter11: Differential Equations

Section11.5: Nonlinear Systems Of Differential Equations

Problem 1YT: YOUR TURN Consider the system of differential equations dx1dt=x1x23x1dx2dt=3x1x26x2 a. Find all...

See similar textbooks

Question

18. Convert the second-order equation
d²y
0
dt2
into a first-order system using v =
dy/dt as usual.
(a) Find the general solution for the dv/dt equation.
(b) Substitute this solution into the dy/dt equation, and find the general solution
of the system.
(c) Sketch the phase portrait of the system.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Trending now

This is a popular solution!

Step by step

Solved in 1 steps with 1 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

YOUR TURN Consider the system of differential equations dx1dt=x1x23x1dx2dt=3x1x26x2 a. Find all equilibrium points. b. Sketch a phase plane diagram, similar to Figure 11.
characterize the equilibrium point for the system x′= Ax and sketch the phase portrait.
The survival of two species depends on their mutual cooperation. For instance, a species of bee that feeds primarily on the nectar of one plant species and simultaneously pollinates the plant. One simple model of mutualism is given by dx/xt = −ax + bxydy/dt = −my + nxy where a, b, m, and n are all positive constants. Perform a graphical analysis and indicate the trajectory directions in the phase plane.
Consider the system dx/dt = -x dy/dt = -4x3+y a. show that the origin is the only equilibrium point. b. find the linearized system at the origin. c. classify the linearized system and sketch its phase portrait.
b) Given the differential equation system: (x' = -7x – 18y = 3x + 8y Find the general solution to the differential equation system. Sketch the phase plane of the differential equation system.
Obtain the phase plane of the system x´= 1 - x2y´ = xy Solve for y=y(x). Please be as clear and legible as possible. Show all the steps. Thank you.
Consider the non-linear autonomous system of differential equations dx dt dy dt =y-1 y x² + 2x - 1 (a) Find the two equilibria of the system. (b) For each equilibrium: i. Use linearisation to classify the equilibrium. ii. Sketch the phase portrait close to that equilibrium. You may find the following results useful: (2D) () = ²() (1) (-4)=-¹(¹) 1) (2), 2
Let the system of differential equations be given by dx 2x(2x – 1) dt dy (x – 1)y dt (a) Find the x-isoclines and y-isoclines. (b) Write down the equilibrium points of the system. (c) Draw the phase diagram for the system. (d) Investigate the stability of the equilibrium points and interpret the phase diagram.
Consider the autonomous system y' = y (y – 3) (a) Write a phase portrait for the above equation. (b) Identify any stable equilibrium solutions. (c) Identify any unstable equilibrium solutions.
Q.1 Equations (1.1) and (1.2) are linear differential equations for a drug delivery system. + 5x - y = 0 Equation (1.1) Equation (1.2) dx dt dy dt - 4x + 2y = 0 - (a) Is the system homogenous or non-homogenous? State your reason. (b) Derive the two-compartment model represented by the differential equations. (c) Construct a general solution (x(t) and y(t)) to the system. (d) Solve for x(t) and y(t) if the initial conditions at t = 0 (s) are x(0)=1 and y(0)=2, respectively.