3. Find all a,b such that all solutions of the differential equation y+ ay+by 0 converge...
i) Find all values of a for which all solutions of the differential equation y/ar 2)y = 0, x0 approach zero as x -+ 0. i) Find all values of a for which all solutions of the differential equation y/ar 2)y = 0, x0 approach zero as x -+ 0.
2. In these problems, determine a differential equation of the form dy/dt = ay+b whose solutions have the required behavior as t →00. Hint: If y=3 is the equilibrium solution, find an equation to relate a and b to each other. There are many answers that satisfy this, but one governing principle that belies them (a) All solutions approach y = 3. (h) All solutions diverge from u = 1/3
Find an autonomous differential equation with all of the following properties: equilibrium solutions at y=0 and y=3, y' > 0 for 0<y<3 and y' < 0 for -inf < y < 0 and 3 < y < inf dy/dx =
Consider the partial differential equation for the function y(x, y) ay ay Әх ду? - ryu = 0 (i) State whether this equation is linear homogeneous, linear inhomogeneous, or non-linear. Justify your statement. (ii) Separate the variables in this equation. Find the separate equations for the variables x and y. (iii) Find the general solutions for each of the separated equations.
D.E. (1) y Find the general solution of the differential equation ay - 25 y' + 25 y = 0. (2) Find the particular solution of the initial-value problem y .+ y - 2 y = 0; y(O) = 5, y (0) - - 1 (3) Find the general solution of the differential equation - NO OVERLAP! y. - 3 y - y + 3 y = 54 x - 3e 2x (4) Find the general solution of the differential...
(a). (3 points) Suppose the solutions of differential equation xy'''−y'' = 0 are in the form of xr where r is some number. Find three solutions in the form of xr. (b). (5 points) Find the general solution of xy'''−y'' = 6x^3
Find the general solution of the given differential equation, and use it to determine how solutions behave as t → 0. y + 7y = t+e-5t QC. 0 Solutions converge to the function y =
Find two power series solutions of the given differential equation about the ordinary point x = 0. y′′ − 4xy′ + y = 0 Find two power series solutions of the given differential equation about the ordinary point x = 0. y!' - 4xy' + y = 0 Step 1 We are asked to find two power series solutions to the following homogenous linear second-order differential equation. y" - 4xy' + y = 0 By Theorem 6.2.1, we know two...
Consider the differential equation y" – 7y + 12 y = 0. (a) Find r1, 72, roots of the characteristic polynomial of the equation above. 11,2 M (b) Find a set of real-valued fundamental solutions to the differential equation above. yı(t) M y2(t) M (C) Find the solution y of the the differential equation above that satisfies the initial conditions y(0) = -4, y'(0) = 1. g(t) = M Consider the differential equation y" – 64 +9y=0. (a) Find r1...
(a) Draw a direction field for the given differential equation. (b) Based on an inspection of the direction field, describe how solutions behave for large t. All solutions seem to approach a line in the region where the negative and positive slopes meet each other. The solutions appear to be oscillatory. All solutions seem to eventually have positive slopes, and hence increase without bound. If y(0) > 0, solutions appear to eventually have positive slopes, and hence increase without bound....