we assume that X solves the differential equation X'=AX.
we assume that X solves the differential equation X'=AX. In this problem, we will investigate the strategy to deal...
In this problem, we will investigate the strategy to deal with repeated eigenvalues in two wavs. Consider A-I 7-2-6 1. Find the only eigenvalue λο of A. Calculate (A-λ01)2 and (A-λ01)3. Does | 0 | satisfy (A-λ01YP P 0 or not? 2. Let M (PIAPIAP). Compute M-1AM. 3. Let Y -M-1X. From the system of equation for Y, deduce the differential equation of order n containing only уз (it should not contain yl or y2) and solve it 4. Obtain...
3. Let L be the linear transformation on R2 that reflects each point P across the line y kx (k> 0) are eigenvectors of L a) (2 marks) Show that v1 and vz b) (1 mark) What is the eigenvalue corresponding to each eigenvector? (Hint: No need to calculate the characteristic polynomial or solve a matrix equation. Geometric reasoning should suffice to solve this problem. Drawing a diagram is recommended!) 3. Let L be the linear transformation on R2 that...
3. Let L be the linear transformation on R2 that reflects each point P across the line y kx (k>0) a) (2 marks) Show that v and v2 - 1 are eigenvectors of L. b) (1 mark) What is the eigenvalue corresponding to each eigenvector? (Hint: No need to calculate the characteristic polynomial or solve a matrix equation. Geometric reasoning should suffice to solve this problem. Drawing a diagram is recommended!) 3. Let L be the linear transformation on R2...
please solve all 3 Differential Equation problems 3.8.7 Question Help Consider the following eigenvalue problem for which all of its eigenvalues are nonnegative y',thy-0; y(0)-0, y(1) + y'(1)-0 (a) Show that λ =0 is not an eigenvalue (b) Show that the eigenfunctions are the functions {sin α11,o, where αη įs the nth positive root of the equation tan z -z (c) Draw a sketch indicating the roots as the points of intersection of the curves y tan z and y...
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...
Question 8 please 5. We start with Schrodinger's Equation in 2(x,t) = H¥(x,t). We can write the time derivative as 2.4(x, t) = V(x,+) - (xt), where At is a sufficiently small increment of time. Plug the algebraic form of the derivative into Schrodinger's Eq. and solve for '(x,t+At). b. Put your answer in the form (x,t+At) = T '(x,t). c. What physically does the operator T do to the function '(x,t)? d. Deduce an expression for '(x,t+24t), in terms...
Discretization, ODE solving, condition number. Consider the differential equation 5y"(x) - 2y'(x) +10y(x)0 on the interval x E [0,10] with boundary conditions y(0)2 and y (10) 3 we set up a finite difference scheme as follows. Divide [0,10] into N-10 sub-intervals, i.e. {xo, X1, [0,1,. 10. Denote xi Xo + ih (here, h- 1) and yi E y(x). Approximate the derivatives as follows X10- 2h we have the following equations representing the ODE at each point Xi ,i = 1,...
Differention Equations - Can someone answer the checked numbers please? Determinants 659 is the characteristic equation of A with λ replaced by /L we can multiply by A-1 to get o get Now solve for A1, noting that ao- det A0 The matrix A-0 22 has characteristic equation 0 0 2 2-A)P-8-12A +62- 0, so 8A1-12+6A -A, r 8A1-12 Hence we need only divide by 8 after computing 6A+. 23 1 4 12 10 4 -64 EXERCISES 1. Find AB,...