[17] For each of the following equations,• find general solutions;• solve the initial value problem with initial condition y(0) = –1, y'(0) = 2;• sketch the phase portrait, identify the type of each equilibrium, and determine thestability of each equilibrium.(a) 2y" + 9y' + 4y = 0(d) 2y" – 3y = 0(g) 9y" + 6y' + y = 0(b) у" + 2у - 8у —D0(е) у"- 2у + 5у 3D0(c) 4y" – 12y + 5y = 0(f) 4y" + 9y = 0-

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Answered: [17] For each of the following…

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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Q4: Sketch the phase plane portrait of equation (without explicitly computing its solutions), x"(t) - 2x' + 2x = 0. What type is it, what happens to solutions as t → ∞o?
please answer part E
(2) Consider the autonomous differential equation y = y4 – ? for a function y = y(t). (a) What are the equilibrium solutions? (b) Sketch the possibility for solutions to this differential equation. (c) Suppose y(0) = 1. Use Euler's method with two increments to estimate the value y(-0.2). Does that answer agree with the sketch in (b)?
[11] For each of the following equations, find general solutions; solve the initial value problem with initial condition x₁ (0) = -1, x₂ (0) = 2; • sketch the phase portrait, identify the type of each equilibrium, and determine the stability of each equilibrium. -2 -13 dx₁/dt = x₁ + x2 x₁ = x₁4x₂ (a) x' = [ X (b) { 4 dx2/dt x₂ = 5x₁²x₂ 1 = -2x1 - x2
Consider the autonomous differential equation. y' = (y – 1)(3 – y)² Sketch solutions curves in the t – y plane corresponding to the following initial conditions. Make sure you label your graph and label any equilibrium solutions. y(0) = 4 y(0) = 3 y(0) = 2 y(0) = 1 y(0) = 0. %3D
find the general solution of y''+y=cos(t) using the undetermined coefficients method
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.
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.
Consider the autonomous differential equation y = (y – 1)2y(y+ 1). (a) Draw a phase line for this differential equation, labeling each equilibrium with its value, and identifying it as a source, sink, or node. (b) On the axes below, provide (rough) sketches for t2 0 of the solutions of the initial value problems corresponding to each of the cases: y(0) = 1, y(0) = 1/4, y(0) = -1/2. 1.5 0.5 3 -0.50 -1.5
2a. Find a change of variable that transforms the equation into an autonomous equation change of variable: new equation: b. Sketch the phase line for the resulting equation and use it to sketch graphs of the long-term behaviors of all the qualitatively different solutions for the new variable, and then for the original equation.
2 Consider the differential equation (t) = x(t) – 4. a.) Find the backward solution of the equation, given an initial value x(0) = 0.5 . b.) ( Is the equation stable or unstable? Why? c.) Sketch the phase diagram and a time path for the equation.
2. [15 points] The following differential equation describes the steady-state concentration of a substance that reacts with first-order kinetics in an axially-dispersed plug-flow reactor, as shown in Figure 1: d² c D dx² dc dx U kc = 0, where D = the dispersion coefficient (m2/h), c = concentration (mol/L), U = the velocity (m/h), and k = the reaction rate (1/h). The boundary conditions can be formulated as Ucin = Uc(x = 0) - D dc (x 0) dx dc (x = L) = 0, dx where Cin = the concentration in the inflow (mol/L), and L = the length of the reactor (m). Use the finite-difference method to solve for concentration as a function of distance, given the following parameters: D = 5000 m²/h, U = 100 m/h, k = 2 1/h, L = 100 m, and cin = 100 mol/L. Use centered finite-difference approximations with Ax = 10 m to obtain your solution. Compare your numerical results with the analytical solution: C= Ucin (U — DÀ1)À2e^2ª — (U — DÀ2)À¹³1 × (+20 +212 - 1+1+2) - 4kD ---/ (+1√17) - 10 (1√+420) U 1 = 2D 4kD U…

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