Sketch the solution curves in the phase plane and the r(t)- and y(t)-graphs for the solutions corresponding to the IVP specified.dr= -2r - 2ydt= -2r + y.dtwhere the initial condition (r(0), y(0)) is (1,0).

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Answered: Sketch the solution curves in the phase…

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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y′=2y−1,y(0)=1
dy The following problem involves an equation of the form = f(y). dt Sketch the graph of f(y) versus y, determine the critical (equilibrium) points, and classify each one as asymptotically stable or unstable. Draw the phase line, and sketch several graphs of solutions in the ty-plane. dy -2y - 1, = e -∞< y0 <∞ dt The function y(t) = 0 is an asymptotically stable equilibrium solution. The function y(t) = 2 is an unstable equilibrium solution. ▼
Name: opez Show work/reasoning. 1. Solve the initial valued problem: y'=e' with y(0) = 0. Give your answer in explicit form y = f(x). X dy y = 2. Solve + dx √x 3. Solve (2+ ye")dx + (xe" -2y)dy = 0 4. Determine if x, x² -1, x² +1 are linearly dependent or independent. 5. Solve y"+2y"-9y'-18y=0
find the general solution of y''+y=cos(t) using the undetermined coefficients method
Construct an autonomous first-order differential equation dy/dx = f(y) whose phase portrait is consistent with the given figure. | (y- 1) (y– 3) (y– 5) dy y 5r 3
y′=−sin y+1,y(0)=0
Consider the pair of differential equations dx dy (4 – x²) (y + 2), dt (x – 1) (3² – 1) . - dt (a) Find all the equilibrium points of these equations. (b) Using Maxima, or otherwise, plot the phase portrait of these equations. (c) Use your phase portrait to state the start and end of the phase path that goes through the origin.
Choose one an unstable equilibrium solution. an asymptotically stable equilibrium solution. no equilibrium solution at all. a semistable equilibrium solution.
4. Consider the autonomous ODE, y' = f(y) = -y(y² – y – 6). (a) Identify the equilibrium solutions and classify the stability of each. d²y directly dt? (b) Identify points of inflection for the solution curves by finding an expression for from the autonomous differential equation. d to the equation dt dy = f(y).) dt (Apply - (c) Sketch a few solution curves in the (t, y) plane, including the location of inflection points.
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Sketch the solution curves in the phase plane and the z(t)- and y(t)-graphs for the solutions corresponding to the IVP specified. dr = -2r - 2y dt = -2r +y. dt where the initial condition (r(0). y(0)) is (1,0).