Find and classify the equilibrium solutions for the each of of the following differentia equations. (a) y = 3-y (b) y = 5+y (c) y'= y²(3-y) (d) y'= y² - 5y + 6 (e) y' = (1 − y)(y — 2) (y − 3) (f) y = y - y³
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- 9. Find a quadratic function y = ao + a1x + a2x² satisfying y(1) = 1, y(-1) = 2, and y'(1) = 2.y′=2x/(1+2y),y(2)=0If y1 is a solution of y" + P1(t)y" + P2(t)y' + P3(t)y = 0, then the substitution y = y1(t)v(t) leads to the following second order equation for v': Yıv" + (3yı' + P1Yı)" + (3y1" + 2p1Yı' + P2Y1)v' = 0. Use the method of reduction of order above to solve the given differential equation. (2 – t)y" + (2t – 3)y" – ty' + y = 0, t< 2; y1(t) = et
- Determine the value of the arbitrary constants y(x) = c₁e³ + c₂е¯³ + 4sin(x) ; y(0) = 1, y'(0) = -1 a. c₁=-2 C₂ = 3 b. c₁=2 C₂ = -3 C. C₁=-2 C2 = 4 d. c₁= -4 C₂ = 3 e. c₁=-2 C₂ = -4Solve the DE. (x³ + xy²-y)dx + (y³ + x²y + x) dy = 0 O arctan(;)=C - ng? (²) = C - x² - y² O 2arctan(;)=C - - O 2arctan O arctan (²) = C - x² - y²Consider the equation y" -y'-2y 0. Assume that y, (t) = e and y2 (1) form a fundamental set of solutions. %3D (b) Let y3 () = - 5e" y() =y (() + 3y, (7) y (1) = 5y, (1) - 3 y (7). Are y, (1), y, (1), and y, (7) also solutions of the givendifferential equation? O Impossible to tell O Yes O No
- Consider the equation y" – 2y' + y = g(t) for some unknown function g(t). Suppose that Y1 and Y2 are two solutions to the equation, and Y1 (0) = 1, Y{(0) Find the function Y – Y2. = 0 and Y2(0) = 2, Y¿(0) = 3.C. Eliminate the arbitrary constants in each equation and express the final answers in the following form: a,n(x)y(n) + an-1(x)yn-1) + . .. + a1(x)y' + ao(x)y – g(x) = 0 . 1. r* – y² = cy 2. y = cje" + cze²" + c3e*r1. Solve the equation y2 + 6x = y(3y – 2x)y'
- Calculate the first, second, and third derivatives of y(t) = 10:3 – 5t2 + 2. (Use symbolic notation and fractions where needed.) y (t) = y"(1) = y"(t) =2.b) Eliminate the Arbitrary Constants in the equation: * x²y = 4 + Cy x?y = 4 + y' x?y" = -4y Ob. a. xy? – 2y' = 0 y = -4y C. d.For the each of the following autonomous equations, find all the equilibrium solutions. For each equilibrium classify whether it is stable, unstable, or semistable. (a) y' = y² — 3y+2 - (b) y' = −y³ +5y² - 6y (c) y' = y(y + 1)² (2 - y) ³ (y² - 4) (d) y'e sin y + 1 - (e) y' = 24 - 2