1a). Consider the equation ay" + by' + cy = d where d ∈ R and a, b and c are positive constants. Show that any solution of this equation approaches d/c as x → +∞. That is, given y(x) a solution we have lim y(x) as x → +∞ = d/c .
1b.) What happens if c = 0?
1c.) What about the case where b = c = 0?
Give a convincing demonstration that the second-order differential equation ay" + by' + cy = 0, where a, b, and c are constants, always possesses at least one solution of the form yı = em where m is a constant and a second solution of the form y, = elo, where k = m is a constant, OR y, = xemx.
If we are given a non-hom. DE. ay''+by'+cy=f. (they are positive constants) 1) Prove that all solutions approach f/c as x approaches infinity? 2) If c=0 what will happen, how will this change the solutions?? 3) If c=0 and b=0 what will happen? Answer all please.
Problem 1: Consider a 2nd order homogeneous differential equation of the form aa2y"(x)bay(x) + cy = 0 (1) where a, b, c are constants satisfy so that y(x) = x (a) Find and justify what conditions should a constant m to (1) is a solution (b) Using your solution to (1) Write these three different cases as an equation that a, b,c satisfy. Hint: Use the quadratic formula we should get three different cases for the values that m can...
Consider the following nonhomogeneous linear differential equation ay 6) + by(s) + cy!4) + dy'"' + ky'' + my' + ny=3x²3x - 7cos +1 where coefficients a, b, c, d, k, m, n are constant. Assume that the general solution of the associated homogeneous linear differential equation is YAEC,+Ce**+ c xe** + c.xe3* + ecos What is the correct form of the particular solution y of given nonhomogeneous linear differential equation? Yanitiniz: o Yo=Ax*e** + Ex + F **+Cxcos() +oxsin()+Ex+F...
8. (10 points) Consider the differential equation (DE) y" + 6y' + cy = 0. where c is some constant and the prime indicates differentiation with respect to t. (i) (2 points) For what value(s) of c does this DE have oscillatory solutions? (ii) (2 points) For what value(s) of c does this DE have an exponentially growing solution? (iii) (3 points) For what value(s) of c does this DE have a constant solution? (iv) (3 points) For what value(s)...
1. (10 points, part I) Consider the following initial boundary value problem lU (la) (1b) (1c) 0L, t> 0 3 cos ( a(x, 0) (a) Classify the partial differential equation (1a) (b) What do the equations (la)-(1c) model? (Hint: Give an interpretation for the PDE, boundary conditions and intial condition.) c) Use the method of separation of variables to separate the above problem into two sub- problems (one that depends on space and the other only on time) (d) What...
Suppose that the auxiliary equation of Ay (6)(x) + By (5) (x) + Cy(4) (x) + Dy'''(x) + Ey''(x) + Fy'(x) + Gy(x) = 0 has solutions of r : = -12 (with multiplicity 3), r = -12+3 i, and r = 13. Select all possible linearly independent solutions of the differential equation. Y = – 2 cos(3t) y = te3t – 12t – 9 sin(3t) = 9t²e = ly 13t 14te' у = y = 11e13t 12t ly =...
= Consider the equation ax² + 2xy + cy? 1 where a > 0 and ac – 62 > 0. Note that a 6 Y = 1. Consider an invertible linear map y X = 6 с х 1 e и between (x, y) and (u, v) given by = y 0 1 V (a) Choose a value for e (in terms of a, b and c), so that the given equation on the (u, v) LEO plane becomes [u...
Consider the matrix transformation T:R → R given by T(x,y,z) = (x+ay, x+(a+1)y, x+ay+z) where a = 13. First use inverse of transformation to find T-(2,1,2). if T-(2,1,2)=(b,c,d) then b+c+d =
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the series converges for any E (-8,s), s > 0 Substitute the series solution y in to the differential equation and simplify the terms to obtain an expression of the form 1-1 where f(r) is a polynomial of degree 2. (c) Determine the values of r....