Concept explainers
Consider the partially decoupled system
(a) Derive the general solution.
(b) Find the equilibrium points of the system.
(c)Find the solution that satisfies the initial condition
(d) Use HPGSystemsolver to plot the phase portrait for this system. Identify the solution curve that corresponds to the solution with initial condition
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Chapter 2 Solutions
Differential Equations
- Determine all critical points of the system of equations- O (0,0) and (-5,6) O(0,0), (0, 1), and (-5,6) ° (0.0), (0.7). and (-5,6) O (1,0) and (-5,6) ° (0.0), (0.2). dx (1, 0), and (-5,6) dt = x - x²- -xy, and dy dt = 7y - xy - 2y².arrow_forwardSolve the systemarrow_forward5. Consider the system <-3-1* -2 x' = x. (a) Find the general solution for the system. (b) Is the equilibrium solution ♂ a source, sink or saddle? Is it unstable or stable? Explain. (c) State the equations of the v- and h-nullclines for the system. (d) Sketch the phase portrait, using nullclines as an aid.arrow_forward
- Please hand write your work if possible or make it clear. I have trouble reading typed answers. Thank you.arrow_forwardFind the general solution of the fourth order linear equation y(4) + 4y" + 4y = 0.arrow_forwardConsider the system of equations (Enter your equation, e.g., y=x.) And for the (non-zero) horizontal (y-)nullcline: (Enter your equation, e.g., y=x.) (Note that there are also nullclines lying along the axes.) dx dt (b) What are the equilibrium points for the system? Equilibria = (Enter the points as comma-separated (x,y) pairs, e.g., (1,2), (3,4).) = x(2 - x - 3y) taking (x, y) > 0. dt dt dt. Recall that a nullcline of this system is a line on which = = 0. Likewise, a vertical nullcline of this system is a line on which = 0, and a dy horizontal nullcline of this system is a line on which = 0. dt (a) Write an equation for the (non-zero) vertical (x-)nullcline of this system: dy dt = y(1-2x), (c) Use your nullclines to estimate trajectories in the phase plane, completing the following sentence: If we start at the initial position (1/2,), trajectories ? ✓the point (Enter the point as an (x,y) pair, e.g., (1,2).)arrow_forward
- Find the general solution of the linear system. Then use the initial conditions to find the particular solution that satisfies them. Use a computer system or graphing calculator to construct a direction field and typical solution curves for the system. x' = 6x + y; y' = - 4x + y; x(0) = 1 y(0) = 0 D. x(t) = C₁e²t+C₂e5t 2t Now find y(t) so that y(t) and the solution for x(t) found in the previous step are a general solution to the system of differential equations. 2t y(t) = -4C₁e²t - ₂5t Find the particular solution. The particular solution is x(t) = C₁ and y(t) =arrow_forwardCorrect solution needed.arrow_forwardDon't give handwritten answer Thanksarrow_forward
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