r' (t)-y-y²-2r y'(l) = -2x
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- (25 points) Find two linearly independent solutions of 2x²y" - xy' + (−4x + 1)y=0, x > 0 of the form Y₁ = ·x³¹(1+α₁x + ª₂x² + a³x³ + ...) Y₂ = x¹² (1+b₁x + b₂x² + b3x³ + ...) where r₁ > 1₂. Enter T1 = a1 = A₂ = a3 T2 = b₁ = b₂ = b3 = || || ||Verify whether y(x) = 2e-x + xe* is a solution of y" + 2y + y = 0.a)Solve the ecuation y''+3y'=12x-5 symbolic b)Solve the initial value problem y''+3y'=12x-5 where y (0) = 1 and y'(0) = 0,symbolic. Call the solution f (x)
- 6) = (" 9) x' -4 a. Find the most general real-valued solution of the linear system 1 ()- ( x(t) = C1 + C2 y(t)nt) Consider the discrete-time dynamical system x+1 = 4x,(1-x₂). 3 If x₁ = ³, X₁+1 Is x₂ = an equilibrium for this system? X₁ 4 Write the updating function rule as f(x) = 4x(1-x). Compute the derivative: f'(x) = Evaluate the derivative at the equilibrium: Is the equilibruim stable, unstable or neither?Find two linearly independent solutions of 2a?y" – xy' + (5x + 1)y= 0, x > 0 of the form Y1 = x" (1+ a1r + a2x? + a3x³+..) Y2 = x" (1+ bịT + bzx² + b3x³+..) where r1 > r2- Enter T1 a1 a2 a3 r2 = %3D b2 b3
- 3 Solve the nonhomogeneous system tY': = 2 (-2) ₁ Y+ (1-2₁).2) Consider the predator-prey model 2 x'= ax -x²-bxy 2 y'=cy - cy² + xy where all the parameters are positive, x is the prey population, y is the predator population and they both grow logistically. Find all equilibria of the model and discuss their stability.Solve the equation xydx - (x + 2y)2dy = 0 ex/y = c y3 (xy) ex/y = c y3 (x-y) ex/y = c y3 (x+y) ex/y = c y4 (x-y)