x = [-5 -8 4 3] x Write the system of ODEs in scalar form using the appropriate

Question

x = [-5 -8 4 3] x Write the system of ODEs in scalar form using the appropriate number of scalar ODEs. Find the general solution of the system. Classify the stability of the equilibrium solution z(t) = (0, 0) and give a qualitative description of all the trajectories of the system, for example, what Is the direction of the trajectories. Explain how you arrived at this description. Draw it qualitative picture of the direction held associated to the system. Solve the initial value problem x(0) = (1, 1). Give a qualitative plot of the solution to the IVP and indicate the direction of the trajectory as t rightarrow infinity

Explanation / Answer

Ans-

Method 1: Substitution. Solve one equation for one variable in terms of the other. then substitute into the other equation. For instance, solving first equation for y : 2y = 37 - 9x y = (37 - 9x)/2 Second eq'n: 5x + 6. (37 - 9x)/2 = 45 5x + 111 - 27x = 45 -22x= -66 x = -66/-22 = 3 ; plug this into expression for y : y = (37 - 9(3))/2 = 5 . Solution: x = 3 , y = 5 . -2: Elimination. Multiply the equations by appropriate constants so that when the equations are added one variable will be eliminated. For instance, to eliminate y . multiply both sides of first equation by -3 : -3. first eq'n: -27x - 6y = -111 second eq'n : 5x + 6v - - Add: -22x = -66 so x = 3. Now sub. x = 3 into one of the original equations, e.g. the second: 5(3) + 6y = 45 so y=5. What we've done geometrically in this example is to find (3,5) as the point of intersection of the lines 9x + 2y = 37 and 5x + 6y = 45 . y'r 2. Systems of nonlinear equations. ExamDle Find the point(s) of intersection of the curves y=3-x2 and y=3-2x. y=3-x2 Set equal to get 3 - x2 = 3 - 2x y=3-2x x2 - 2x = 0 x(x-2) = 0 x= Oor2.

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