Question

Find the general solution to the following linear system around the equilibrium point:

x '1 = 2x1 + x2

x '2 = −x1 + x2

(b) If the initial conditions are x1(0) = 1 and x2(0) = 1, find the exact solutions for x1 and x2.

(c) Plot the exact vector field (precise amplitude of the vector) for at least 4 points around the equilibrium, including the initial condition.

(d) Plot the solution curve starting from the initial condition x1(0) = 1 and x2(0) = 1 on the phase plane.

Answer #1

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Find the solution to the following system by converting the
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x'2= x1-x2, x2(0)=-2

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dx1/dt = 4x1 + 5x2
dx2/dt = x1 + 8x2
, with initial conditions x1(0) = 6, and
x2(0) = 0.

Linear Algebra
find all the solutions of the linear system using Gaussian
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x1-x2+3x3+2x4=1
-x1+x2-2x3+x4=-2
2x1-2x2+7x3+7x4=1

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homogeneous linear systems. Show all algebraic steps.
a. x1 + x2 + x3 = 0.
x1 - x2 - x3 = 0
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x1 - 2x2 + 2x3 + x4 = 0.
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1.) either solve the given system of equations, or else show
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x1 + 2x2 - x3 = 2
2x1 + x2 + x3 = 1
x1 - x2 + 2x3 = -1
2.) determine whether the members of the given set of vectors
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linear relation among them.
(a.) x(1) = (1, 1, 0) , x(2) = (0, 1, 1) ,
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elimination method. (If there is no solution, enter NO SOLUTION. If
there are infinitely many solutions, express your answer in terms
of the parameters t and/or s.)
x1
+
2x2
+
8x3
=
6
x1
+
x2
+
4x3
=
3
(x1,
x2, x3)
=
2)Solve the system of linear equations, using the Gauss-Jordan
elimination method. (If there is no solution, enter NO SOLUTION. If
there are infinitely many solutions, express...

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[x′1 x′2]=[−13 20−6 9][x1
x2] satisfying the initial conditions
[x1(0)x2(0)]=[6−9].
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{x′ = 6x + 4y
{y′=−2x
satisfying the initial conditions x(0)=−5 and
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x(t) = _____
y(t) = _____

Find the solution to the linear system of differential
equations: x'= -19x+30y y'= 10x+16y Satisfying the initial
conditions: x(0)= -7 y(0)= -5

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