d2 a. Find all rcal cigenvalucs and cigenfunctions of the lincar operator b. Find all real cigenvalues and cigenfunctio...
Find the eigenvalues and eigenfunctions for the differential operator L(y)=−y″L(y)=−y″ with boundary conditions y′(0)=0y′(0)=0 and y′(3)=0y′(3)=0, which is equivalent to the following BVP y″+λy=0,y′(0)=0,y′(3)=0.y″+λy=0,y′(0)=0,y′(3)=0. Find the eigenvalues and eigenfunctions for the differential operator L(y)--y" with boundary conditions y (0)0 and y' (3)-0, which is equivalent to the following BVP (a) Find all eigenvalues 2n as function of a positive integer n > 1. (b) Find the eigenfunctions yn corresponding to the eigenvaluesn found in part (a). Help Entering Answers ew...
Please answer C 3. (8 marks total) Show which of the following mappings between real vector spaces are lincar and which are not lincar (a) LRR2 defined by L1(x) (r, 2x). (b) L2 R2 -R2, defined by L2(r, y) (cos(30) -ysin(30), z sin(30) +ycos(30)). (c)L:F(R;R) >R, defined by L()-s()(1) (d) L4 : Cao(R: R) > R, defined by Ldf) =おf(t)dt. (Notes: (i) The real vector space (F(R:R),+) consists of all functions from R to R (i.c. all real-valued functions of...
Please answer D 3. (8 marks total) Show which of the following mappings between real vector spaces are lincar and which are not lincar (a) LRR2 defined by L1(x) (r, 2x). (b) L2 R2 -R2, defined by L2(r, y) (cos(30) -ysin(30), z sin(30) +ycos(30)). (c)L:F(R;R) >R, defined by L()-s()(1) (d) L4 : Cao(R: R) > R, defined by Ldf) =おf(t)dt. (Notes: (i) The real vector space (F(R:R),+) consists of all functions from R to R (i.c. all real-valued functions of...
Given y'"- y" - 4y'- 6y=0 (1) , identify Differential Operator L of (1). OL=D3-D2 - 4D - 6 o L = (D - 3)(D2 + 2D + 2) O Both of them are correct! None of them
(*) Find the cigenvalues and eigenfunctions for the following problem: - = A, 0 < x <l, y(0) = 2y(1), y'(0) =y'l), where I > 0 is is a parameter.
If T is a bounded operator on H with one-dimensional there exist vectors y, z E H such that Tx = (x, z)y for all show the following: sional range, show tha x H. Hence 0 (b) T-AT, λ is a scalar. If T is a bounded operator on H with one-dimensional there exist vectors y, z E H such that Tx = (x, z)y for all show the following: sional range, show tha x H. Hence 0 (b) T-AT,...
QUESTION 11 Find a differential operator that annihilates the given function. xºe-3xsin (-12x) +X A[(D+3) 2+144]+D2 os (0+3)2+144]10 ocllo-3)2 +144]10 ool(D-3)2-144] 1070 Oe [(0+3)2-144]10
Find linearly independent functions that are annihilated by the given differential operator. (Give as many functions as possible. Use x as the independent variable. Enter your answers as a comma-separated list.) 1. D4 2. D2 − 7D − 44 Solve the given initial-value problem: 2. y'' + y = 10 cos 2x − 4 sin x, y(π/2) = −1, y'(π/2) = 0 : y(x)=____________
Let V be a real vector space. Recall the following definition Definition 1. An operator L : V → V is linear if L(ax + by) = aLx + bLy, Va, b E R, and Vx, y E V. If (1) does not hold, L is called nonlinear. Let VC(-o0, 0o). Check directly using the Definition 1 above if L defined below are linear or nonlinear b) L-돐 +1n 1.1 (Note, this means Ly-y't in ly|.) c) L = a,...
3. Find all a,b such that all solutions of the differential equation y+ ay+by 0 converge to zero asgoes to