2/ Draw the phase portrait and determine the stability of critical point: ✗ 00 +2X°-x²+1=0
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- Determine which of the following pairs of functions are linearly independent. NO ANSWER + 1. f(x) = x² , g(x) = 4|x|2 NO ANSWER * 2. f(1) = ed"cos(Ht) = ed'sin(ut) ,u # 0 NO ANSWER 3. f(0) = cos(30) g(6) = 2 cos (8) – 2 cos(8) %3D NO ANSWER + 4. f(t) = 3t g(t) = |t|Question 8 :SHOW your work. Consider an autonomous differential equation -f(y) for which the graph of f(y) vs y is shown below. (20) -1 a) List the equilibrium solutions of this dt (4-16). (2,4) (4:0)Solve (yz + xyz) dx + (zx + xyz) dy (xy + xyz) dz = 0 +
- T/2 V1+cosx dx T/34. A car supported by a MacPherson strut (shock absorber system) travels on a bumpy road at a constant velocity v. The equation modeling the motion of the car is Tut 80x + 1000x = 2500 cos where r = x (t) represents the vertical position of the cars axle relative to its equilib- rium position, and the basic units of measurement are feet and feet per second (this is actually just an example of a forced, un-damped harmonic oscillator, if that is any help). The constant numbers above are related to the characteristics of the car and the strut. Note that the coefficient of time t (inside the cosine) in the forcing term on the right hand side is a frequency, which in this case is directly proportional to the velocity v of the car. (a) Find the general solution to this nonhomogeneous ODE. Note that your answer will have a term in it which is a function of v.Use the method of undetermined coefficients to solve the ODEy′′ − 3y′ + 2y = e^x