Same question both picture Consider the system of equations10(20x'1)xX-520 -10(a) Use a graphing utility to graph a direction field and a fewtrajectories.(b) Describe how the solutions behave as t → ∞.The equilibrium points lie on the line [Choose one ▾Solutions starting at all other pointsX= Choose one ▼(c) Find the general solution of the systerChoose onex2 = -X1x2 = -2x1x2 = x1-2X2 = x1= 2x1x2 = 1 + 2x2 = (a) Use a graphing utility to graph a direction field and a fewtrajectories.(b) Describe how the solutions behave as t→∞0.The equilibrium points lie on the line Choose oneSolutChoose one(c) Pind (50)+[(5)+(9)]10X =C1a(1) + a[(1) + (-₁)]etteC20a(-1/2) + a[(-1/2)¹+ (2)]tC1(1)¹+ 1¹ + 0₂ [ ( 2₂ ) te ¹ + ( ₁ ) e +¹]e2a()+ a[()+(-)]t227ns.

Given, An ordinary differential equation in matrix form - x'=10-520-10x We need to graph the…

Answered: Consider the system of equations 10 -5…

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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Generate a slope field in that shows the solution curve in the RF-plane (rabbit/fox) for the system of predator-prey equations with initial conditions R(0)=8 and F(0)=1
Use field II to sketch the graphs of the solutions that satisfythe given initial conditions. a) y(0) = 1 b) y(0) = 2 c)y(0) = -1
Find the general solution in terms of real functions. (b) From the roots of the characteristic equation, determine whether each critical point of the corresponding dynamical system is asymptotically stable, stable, or unstable, and classify it as to type. (c) Use the general solution obtained in part (a) to find a two-parameter family of trajectories x=x1i+x2j=yi+y′j of the corresponding dynamical system. Then sketch by hand, or use a computer, to draw a phase portrait, including any straight-line orbits, from this family of trajectories.
Use the direction field to sketch the graphs of the solutions that satisfy the given initial conditions. (a) y(0) = 1(b) y(0) = 2(c) y(0) = –1
(a) Find a slope field whose integral curve through (1, 1) satisfies xy³ −x²y = 0 by differentiating this equation implicitly. (b) Prove that if y(x) is any integral curve of the slope field in part (a), then x[y(x)]³ −x² y(x) will be a constant function. (c) Find an equation that implicitly defines the integral curve through (−1,−1) of the slope field in part (a).
In the study of a nonlinear spring with periodic forcing, the equation y''+ky+ry3= Acos(ωt) arises. Let k = 4, r = 8, A = 1, and ω = 6. Find the first three nonzero terms in the Taylor polynomial approximation to the solution with initial values y(0) = 0, y'(0)=1. The Taylor approximation to three nonzero terms is y(t) =?
Suppose that y' = f(y) has no constant solutions, where f : R -> R is infinitely many times differentiable at all points in R. What is the behavior of solutions to y' = f(y).
Find the linearization L(x) of y=e10xln(x) at a=1
4 a. Consider the i.v.p x' = t^(2) + cos(x), x(0) = 0. Verify that the hypothesis of Cauchy Picard theorem for a suitable domain D. b. Then estimate the interval of existence of the solution.
Sketch the slope field for dy/dt=y2-4. How do the solutions behave as t approaches infinity? Is end behavior dependent on initial conditions? If so, explain how it is dependent.
(a) Solve the initial value problem in explicit form: t ∂u ∂t + (t + u) ∂u ∂x = x − t, u(x, 1) = 1 + x, |x| ≤ 2, sketch typical base characteristics in the xt-plane. Show the subset of R2 in which the solution u(x, t) is defined.
the relationship between aphids, A, ( prey) and ladybugs, L, (preditor) can be described as follows: dA/dt=2A=0.01AL dL/dt=-0.5L+0.0001AL a) find the two critical points of the predator-prey equations b) use the chain rule to write dL.dA in terms of L and A c) suppose that at time t=0, there are 1000 aphids and 200 ladybugs, use the Bluffton university slope field generator to graph the slope for the system and the solution curve. let 0 be less than or equal to A who is less than or equal to 15000 and 0 is less than or equal to L which is less than or equal to 400.

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