1. Consider the linear homogeneous system X'(t) = A(t)X(t) for t >1 with(2t2 +13t3 – t1- t2A =11t21t - t3(a) Verify thatP1(t) =andÞ2(t)are solutions to X'(t) = A(t)X(t).%3D(b) Calculate the Wronskian of {P1, P2}. Are , and 2 linearly independent?(c) Find a fundamental matrix of the system X'(t) = A(t)X(t).%3D(d) Find the solution to the IVPX'(t) = A(t)X(t),X(2)

Given linear homogeneous system X't=AtXt for t>1 with A=2t2+1t3-t31-t21t2-11t-t3⇒A=2t2+1-t…

Answered: 1. Consider the linear homogeneous…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 69EQ: Let x=x(t) be a twice-differentiable function and consider the second order differential equation...

See similar textbooks

Similar questions

Let x=x(t) be a twice-differentiable function and consider the second order differential equation x+ax+bx=0(11) Show that the change of variables y = x' and z = x allows Equation (11) to be written as a system of two linear differential equations in y and z. Show that the characteristic equation of the system in part (a) is 2+a+b=0.
help asap
Find two linearly independent solutions of 2a?y" – xy' + (5x + 1)y= 0, x > 0 of the form Y1 = x" (1+ a1r + a2x? + a3x³+..) Y2 = x" (1+ bịT + bzx² + b3x³+..) where r1 > r2- Enter T1 a1 a2 a3 r2 = %3D b2 b3
of the form y₁ = (1+₁+ a₂²+az³ + ...) 3₂ = x2(1+b₁x + b₂x² + b₂x³ + ...) where T₁ > T2- Enter Find two linearly independent solutions of 2x²y" - xy + (2x+1)y=0, z>0 T1 1 01 = a₂= 03 = T2= 1/2 b₁ = b₂ = b3 =
Find two linearly independent solutions of 2x2y"-xy'+(-6x+1)y=0, x>0 of the form
5. Let h Ah x = [h₁ + ²√5 h = √5], where h is a real parameter. (a) Find all values of h for which the system X' = AX has either a center, a spiral sink, or a spiral source. (b) Find all values of h for which X' = AX has a sink or a source.
Calculate and simplify the Wronskian of the following two functions: y₁ (t) = 3t+3, y2(t) = 1² + 1 Based on the value of the Wronskian, are y₁ and y2 linearly dependent or linearly independent?
and x® (t) = (). t (2) Consider the vectors x(t) = ( 10) (a) Compute the Wronskian of x(1) and x(2). X W (b) In what intervals are x1) and x2) linearly independent? D (c) What conclusion can be drawn about coefficients in the system of homogeneous differential equations satisfied by x1) and x2)? Choose one ▼ of the coefficients of the ODE in standard form must be discontinuous at to (d) Find the system of equations x' = P(t)x. P(t) =
= 12. (a) Find the unique quadratic polynomial y ax²+bx+c that passes through the three points (1, 1), (3,5), and (-2, 0) in the x-y plane. (Hint: this will involve a linear system where a, b, and c are the unknowns.) (b) How many data points (x, y) should be needed to uniquely determine the coefficients of an nth degree polynomial anx" +an-1² -1xn-1+...+ a₁x+ao? Explain your answer in terms of linear systems.
Consider the ecuation ky" – ky' – 210y = 0, with constant real k + 0. If one - %3D solution of the ecuation is: Y1(x) = e then another solution linearly %3D dependent of the equation with y1 is: O a) y2(z) = el7z O b) y2(x) = ekz Oc) y2(x) = e¬1llz %3D O d) y2(x) = e-5z %3D
(B. Janssen, KTH, 2014) Consider the linear system 0.550x+0.423y = 0.127 0.484x + 0.372y = 0.112 Suppose we are given two possible solutions, u = [_11] and v- -1.91. 1.01 0.9 a. Decide based on the residuals b - Au and b - Av which of the two possible solutions is the 'better' solution. b. Calculate the exact solution x. c. Compute the errors to the exact solution. That is, compute the infinity norms of u-x and v-x. Do the results change your answer to 7a?
Find a particular solution of the indicated linear system that satisfies the initial conditions x₁ (0) = 7, x₂(0) = -3. 1 5t [240] X': 7 - 2 12 -7 5t *2=6²08 ... The particular solution is x₁ (t)= and x₂ (t) = 1 6

SEE MORE QUESTIONS

Recommended textbooks for you

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS

1. Consider the linear homogeneous system X'(t) = A(t)X(t) for t >1 with 2t2 +1 3 t3 – t 1- t2 A = 1 1 t - t3 (a) Verify that P1(t) = () 2(t) = and are solutions to X'(t) = A(t)X(t). %3D (b) Calculate the Wronskian of {P1, Þ2}. Are , and 2 linearly independent?