Given X′=AX with X(t)=[x(t)y(t)], A=[23−4012−21] and X(0)=[3−4]

Given X′=AX with X(t)=[x(t)y(t)], A=[23−4012−21] and X(0)=[3−4]. (a) Write the eigenvalues and eigenvectors of A λ1= , V1= ⎡⎣⎢⎢ ⎤⎦⎥⎥ , and λ2= , V2= ⎡⎣⎢⎢ ⎤⎦⎥⎥ (b) Write the solution of the initial-value problem in terms of x(t),y(t) x(t)= y(t)=

Expert Solution

Colby Messinger answered 2 years ago

Given the two lines (x, y, z) = (4, -3, 5) + t(2, 0, -3) (x,...

Given the two lines (x, y, z) = (4, -3, 5) + t(2, 0, -3) (x, y, z) = (4, -3, 5) + s(5, 1, -1) a) determine a vector equation for the plane that combines the two lines b) parametric equations of the plane that contains the two lines c) Cartesian equation of the plane that contains the two lines

Consider the initial value problem X′=AX, X(0)=[−4,-2], with A=[−6,0,1,−6] and X=[x(t)y(t)] (a) Find the eigenvalue λ,...

Consider the initial value problem X′=AX, X(0)=[−4,-2], with A=[−6,0,1,−6] and X=[x(t)y(t)] (a) Find the eigenvalue λ, an eigenvector V1, and a generalized eigenvector V2 for the coefficient matrix of this linear system. λ= , V1= ⎡⎣⎢⎢ ⎤⎦⎥⎥ , V2= ⎡⎣⎢⎢ ⎤⎦⎥⎥ $ (b) Find the most general real-valued solution to the linear system of differential equations. Use t as the independent variable in your answers. X(t)=c1 ⎡⎣⎢⎢ ⎤⎦⎥⎥ + c2 ⎡⎣⎢⎢ ⎤⎦⎥⎥ (c) Solve the original initial value problem. x(t)=...

y''(t)+(x+y)^2y(t)=sin(xt+yt)-sin(xt-y*t), y(0)=0, y'(0)=0, x and y are real numbers

y''(t)+(x+y)^2*y(t)=sin(x*t+y*t)-sin(x*t-y*t), y(0)=0, y'(0)=0, x and y are real numbers

Solve the following differential equations: 1. x"(t)+ x(t)=6sin(2t) ; x(0)=3, x'(0)=1 2.y"(t)- y(t)=4cos(t) ; y(0)+0 ,...

Solve the following differential equations: 1. x"(t)+ x(t)=6sin(2t) ; x(0)=3, x'(0)=1 2.y"(t)- y(t)=4cos(t) ; y(0)+0 , y'(0)=1

if y(t) is the solution of y′′+2y′+y=δ(t−3),y(0)=0,y′(0)=0 the find y(4)

Solve the following differential equations: 1.) y"(t)- 6 y'(t)+9 y(t)=6t^2e^(3t) ; y(0)=y'(0)=0 2.)x"(t)+4x(t)=t +4 ; x(0)=1...

Solve the following differential equations: 1.) y"(t)- 6 y'(t)+9 y(t)=6t^2e^(3t) ; y(0)=y'(0)=0 2.)x"(t)+4x(t)=t +4 ; x(0)=1 , x'(0)=0

Solve the following differential equations: 1.) y"(x)+ y(x)=4e^x ; y(0)=0, y'(0)=0 2.) x"(t)+3x'(t)+2x(t)=4t^2 ; x(0)=0, x'(0)=0

in the problem given (3 - x)y" + x2y' + (1 + x)y = 0 ,...

in the problem given (3 - x)y" + x2y' + (1 + x)y = 0 , x0=0. find the recurrence relation and the series solution given y(0) = -6 and y'(0)= 2

Solve this differential equation y''+(-4-2-2)y'+(4+4+4+4)y=x y(0)=3-2 y'(0)=2-3 Answer it as y(x)=... and motivate all the steps...

Solve this differential equation y''+(-4-2-2)y'+(4+4+4+4)y=x y(0)=3-2 y'(0)=2-3 Answer it as y(x)=... and motivate all the steps of the calculation

(8) Suppose T : R 4 → R 4 with T(x) = Ax is a linear...

(8) Suppose T : R 4 → R 4 with T(x) = Ax is a linear transformation such that • (0, 0, 1, 0) and (0, 0, 0, 1) lie in the kernel of T, and • all vectors of the form (x1, x2, 0, 0) are reflected about the line 2x1 − x2 = 0. (a) Compute all the eigenvalues of A and a basis of each eigenspace. (b) Is A invertible? Explain. (c) Is A diagonalizable? If yes,...

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