Thank you
1)Determine if the following functions can be wronskians in (-1,1) of two solutions of some homogeneous...
2. (Sturm-Liouville Theory) Consider the following linear homogeneous second-order differential equation and boundary conditions v(T where a and b are finite, p(x), p(x,)) are real and continuous on [a, b), and p(x),w(x) > 0 on a,b]. Show that two distinct solutions to this ODE, Pm(z) and (x), are orthogonal to each other on the interval [a,b]. That is, prove the following relationship 0 2. (Sturm-Liouville Theory) Consider the following linear homogeneous second-order differential equation and boundary conditions v(T where a...
3. Two solutions of the following linear equation system are x1, X2, where Xi = (1,1,-3,1), x2-x1 + xd xd that makes cTx2 - cTx1 - 1, where c [1 1 2 1] Find every Ax=11 2 2 3 |x=b 3. Two solutions of the following linear equation system are x1, X2, where Xi = (1,1,-3,1), x2-x1 + xd xd that makes cTx2 - cTx1 - 1, where c [1 1 2 1] Find every Ax=11 2 2 3 |x=b
The indicated functions are known linearly independent solutions of the associated homogeneous differential equation on (0, 0). Find the general solution of the given nonhomogeneous equation. *?y" + xy' + (x2 - 1)y = x3/2; Y1 = x-1/2 cos(x), Y2 = x-1/2 sin(x) y(x) =
2. You can use Dand write an operator instead of an equation in this question. (a) Find a constant coefficient linear homogeneous differential equation of lowest order that has n(x)-x , y2(z) = x2 , and y3(z) = eェamong its solutions. (b) Now find a different linear homogeneous differential equation of an order lower than the one in (a) that has the same y1,U2,U3 among its solutions. (c) Find a constant coefficient linear homogeneous differential equation of lowest order that...
Exercise 5.27 Suppose and 2t) are solutions of a linear homogeneous system A (t)x with a coefficient matrix A(t) that is continuous on an interval a < t < β. Prove that the determinant s() -det( 3) (t) 2 is either never equal to 0 for α < t < β or else it is identically (i.e., alu ays) equal to 0 on α < t < β. (Hint: by direct calculation show the determinant satisfies the first order, linear...
If the functions y = x and y = xe" are linearly independent solutions of the non-homogeneous second-order linear differential equation with variable coefficients z_ yll – x(x + 2)y + (x+2)y=r, its general solution is given by Oy=C1 + C2xe" + x2 O y=C1x + C2xe - 22 None of them O y=C12 + C2z²er - 23 Oy=C12? + Cymet – x3
Let f(x) and g(x) be any two functions from the vector space, C[-1,1] (the set of all continuous functions defined on the closed interval [-1,1]). Define the inner product <f(x), g(x) >= x)g(x) dx Find <f(x), g(x) > when f(x) = 1 – x2 and g(x) = x - 1
just focus on A,B,D 1. Homogeneous ODE Find a general solution of the linear non-constant coefficient, homogeneous ODE for y(x) x3y'" – 3xy" + (6 – x2)xy' – (6 – x?)y = 0 as follows. a) You are given that yı(x) = x is a solution to the above homogeneous ODE. Confirm (by substitution) that this is the case. b) Apply reduction of order to find the remaining two solutions, then state the general solution. (Hint: The substitution y2(x) =...
1. (a) Determine the largest x-interval where the initial value problem has a unique solution: 1 1 (22 – 40) (6) + y(5) + (x + 1)y" + e*y' + (tan x)y In (x – 1) x2 9 = = = with y(2.5) A, Y' (2.5) B, y" (2.5) C, y'" (2.5) D, y(4) (2.5) y(5)(2.5) = F, where A, B, C, D, E, and F are some known constants. E, (b) Determine whether the set of functions {5, sin’...
(1 point) The general solution of the homogeneous differential equation can be written as 2 where a, b are arbitrary constants and is a particular solution of the nonhomogeneous equation By superposition, the general solution of the equation 2y 5ryy 18z+1 isyp so yax-1+bx-5+1+3x NOTE: you must use a, b for the arbitrary constants. Find the solution satisfying the initial conditions y(1) 3, y'(1) 8 The fundamental theorem for linear IVPs shows that this solution is the unique solution to...