For equilibrium solutions, put y'=0
Then, (y+1)(y-1)arctan(y)=0
Which implies y+1=0,y-1=0, arctan (y) =0
Thus, equilibrium solutions are y=-1,1 and 0
(c) List the equilibrium solutions of the differential equation y' = (y2 – 1) arctan(y)
2. Differential equations and direction fields (a) Find the general solution to the differential equation y' = 20e3+ + + (b) Find the particular solution to the initial value problem y' = 64 – 102, y(0) = 11. (e) List the equilibrium solutions of the differential equation V = (y2 - 1) arctan() (d) List all equilibrium solutions of the differential equation, and classify the stability of each: V = y(y - 6)(n-10) (e) Use equilibrium solutions and stability analysis...
Bonus (Abel's formula) a) Show that if y1 and y2 are solutions to the differential equation y"p(t)y(t)y 0 where p and q are continuous on an interval I, then the Wronskian of y and y2, W(y1,y2) (t) is given by - Sp(t)dt ce W(y1, y2)(t) where c depends on y and y2 (b) Use Abel's formula to find the Wronskian of two solutions to the differential equation ty"(t 1)y 3y 0 Do not solve the differential equation
a) Assume that y1(c) t and y2)te are solutions of the differential equation t2y_ t(t + 2))" + t(t + 2)y-0, t > 0 Do y1(t) and y2() form a fundamental set of solutions of the O.D.E.? C) State the general solution for this O.D.E. a) Assume that y1(c) t and y2)te are solutions of the differential equation t2y_ t(t + 2))" + t(t + 2)y-0, t > 0 Do y1(t) and y2() form a fundamental set of solutions of...
true or false If yı(t) and y2(t) are two solutions of the differential equation y2 – y' +y = 0, then for any constants cı and c2, cıyı(t) + C2y2(t) is also a solution. Doğru Yanlış
if y1(t) and y2(t) are two solutions of the differential equation y^2-y'+y=0 then for any constants c1 and c2 c1y1(t)+c2y2(t) is also a solution true or false and why
Consider the differential equation y' (t) = (y-4)(1 + y). a) Find the solutions that are constant, for all t2 0 (the equilibrium solutions). b) In what regions are solutions increasing? Decreasing? c) Which initial conditions y(0) = A lead to solutions that are increasing in time? Decreasing? d) Sketch the direction field and verify that it is consistent with parts a through c. a) The solutions are constant for (Type an equation. Use a comma to separate answers as...
Three linearly independent solutions of the differential equation y'"' - y" - 6y' = 0 are Select the correct answer. a. V1 =e-6s, y2 =xe-1, V3 =1 b. Y1 = 224, y2 = 2-3x, y3 = 1 c. Y1 = 2-6x, y2 = e", y3 = 1 d. Y1 = e3x, y2 = 2-2*, y3 = 1 e. Vi=e , y2=xe-1, V3=1
(4 points) Write down a first order autonomous differential equation which has equilibrium solutions at y = 1, y = 2 and y = 3. Is the differential equation linear?
Suppose you are told that a certain differential equation has the following equilibrium solutions: y 4.01726 y=6 y=8 If the initial condition for the differential equation is y(0) = 5.88269, which of the following are possible values for y(8.62729)?
6. Consider the autonomous differential equation (a) Find all of its equilibrium solutions. (b) Classify the stability of each equilibrium solution. Justify your answer. (c) If y(t) is a solution that satisfies y(-1) =-4, what is y(0)? Without solving the equation, briefly explain your conclusion. (d) If y() is a solution that satisis y(3) -3, then what is lim y(t)? 6. Consider the autonomous differential equation (a) Find all of its equilibrium solutions. (b) Classify the stability of each equilibrium...