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7. Consider the one-parameter family of linear systems depending on the parameter a given by X,#A...
please explain everything! 7. Consider the one-parameter family of linear systems depending on the parameter a...
please explain
everything!
7. Consider the one-parameter family of linear systems depending on the parameter a given by X,#AX with A-|-1 İa] This family can be represented in the trace-determinant plane as a curve. -1 0 a. Sketch the curve representing this family in the trace-determinant plane. We were unable to transcribe this image
1. Construct a table of the possible linear systems as follows: (a) The first column contains the...
1 and 2
1. Construct a table of the possible linear systems as follows: (a) The first column contains the type of the system (sink, spiral sink, source, if it has a name. (b) The second column contains the condition on the eigenvalues that corresponds (c) The third column contains a small picture of two or more possible phase por- (d) The fourth column contains x(1)-and y(a)-graphs of typical solutions indicated Hint: The most complete table contains 14 cases. Don't...
Consider the linear system X' = AX where A is defined by , where a and...
Consider the linear system X' = AX where A is defined by
,
where a and b are real numbers. Assume that the determinant of A
is not zero.
Classify the equilibrium solution (0,0) depending on the signs
of a and b. Also, sketch a few trajectories for each case,
including a few tangent vectors.
3. Homogeneous linear systems with complex and repeated eigenvalues. Find the general solu- tion of the given system of...
3. Homogeneous linear systems with complex and repeated eigenvalues. Find the general solu- tion of the given system of differential equations. For the two-dimensional systems, classify the origin in terms of stability and sketch the phase plane (a) x'(t) y'(t) 6х — у, 5х + 2y. = (b) 4 -5 x'(i) х. -4 (c) 1 -1 2 x'() -1 1 0x -1 0 1
3. Homogeneous linear systems with complex and repeated eigenvalues. Find the general solu- tion of the...
Problem 1 (Linear Systems of Equations). (a) Determine the values of a for which the follow- ing ...
Problem 1 (Linear Systems of Equations). (a) Determine the values of a for which the follow- ing system of equations have no solution, exactly one solution, infinitely many solutions (a + 2)y + (a2-4)2 = (0-2) (b) If A = 4-1 0 a 2b a a be the augmented matrix of a linear system of equations then evaluate the values of a and b for which the linear system has no solution? exactly one solution? one parameter solution? two parameter...
(6) (Hand in!) Consider the linear function L : R3 → R3, given by the formula L(2)-Ax, where A is...
(6) (Hand in!) Consider the linear function L : R3 → R3, given by the formula L(2)-Ax, where A is one of the following 3x 3 matrices: 0 -1 3 0 -21 0 1-2 3 1 -3 -1 2 0, 1 -1 1 21-3 Compute the determinant of each matrix. In each case what information does the determinant give you regarding: (i.) the invertibility of the function of L, (ii.) the volume of the parallelogram formed by passing the standard...
Consider the linear system of first order differential equations x' = Ax, where x= x(t), t...
Consider the linear system of first order differential equations x' = Ax, where x= x(t), t > 0, and A has the eigenvalues and eigenvectors below. 4 2 11 = -2, V1 = 2 0 3 12 = -3, V2= 13 = -3, V3 = 1 7 2 i) Identify three solutions to the system, xi(t), xz(t), and x3(t). ii) Use a determinant to identify values of t, if any, where X1, X2, and x3 form a fundamental set of...
Section B - Answer any two questions. 2. (a) Consider the one-parameter family of nonlinear ordin...
Section B - Answer any two questions. 2. (a) Consider the one-parameter family of nonlinear ordinary differential equations dr where a is a real parameter. i. Find all equilibrium points. ii. Sketch the bifurcation diagram, and indicate the behaviour of non- equilibrium solutions using appropriate arrows. ii. Find all bifurcation points and classify them. 10 Marks (b) Consider the second order differential equation i. Show that (1) can be written as the system of ordinary differential equations (y R for...
Consider the equation 1. Find the trace ? and the determinant ?. Sketch the graphs of...
Consider the equation
1. Find the trace ? and the determinant ?. Sketch the graphs of
? vs. b and ? vs. b. Then sketch the graph of the curve ? = ? (b),
? = ?(b) in the ??-plane along with ?^2 = 4?, ? = 0 and ? = 0(? ?
0).
2. Find the intersection points of the the curve ? = ?(b),? =
?(b) with ?^2 = 4?, ? = 0 and ? = 0(? ?...
7. (14,5) A particular parametric curve is given by x=1-3, y = 21 +3 for -2...
7. (14,5) A particular parametric curve is given by x=1-3, y = 21 +3 for -2 515 3. Sketch the curve using arrows to indicate the direction in which the curve is traced as increases. Then eliminate the parameter / to find a Cartesian equation of the curve. 8. (10,4) Find the equation of the plane through the point (-5, 4, 2) and with normal vector-31 +4j - k. Give your answer in both the vector equation of a plane...
please explain
everything!
7. Consider the one-parameter family of linear systems depending on the parameter a given by X,#AX with A-|-1 İa] This family can be represented in the trace-determinant plane as a curve. -1 0 a. Sketch the curve representing this family in the trace-determinant plane. We were unable to transcribe this image
1 and 2
1. Construct a table of the possible linear systems as follows: (a) The first column contains the type of the system (sink, spiral sink, source, if it has a name. (b) The second column contains the condition on the eigenvalues that corresponds (c) The third column contains a small picture of two or more possible phase por- (d) The fourth column contains x(1)-and y(a)-graphs of typical solutions indicated Hint: The most complete table contains 14 cases. Don't...
Consider the linear system X' = AX where A is defined by
,
where a and b are real numbers. Assume that the determinant of A
is not zero.
Classify the equilibrium solution (0,0) depending on the signs
of a and b. Also, sketch a few trajectories for each case,
including a few tangent vectors.
3. Homogeneous linear systems with complex and repeated eigenvalues. Find the general solu- tion of the given system of differential equations. For the two-dimensional systems, classify the origin in terms of stability and sketch the phase plane (a) x'(t) y'(t) 6х — у, 5х + 2y. = (b) 4 -5 x'(i) х. -4 (c) 1 -1 2 x'() -1 1 0x -1 0 1
3. Homogeneous linear systems with complex and repeated eigenvalues. Find the general solu- tion of the...
Problem 1 (Linear Systems of Equations). (a) Determine the values of a for which the follow- ing system of equations have no solution, exactly one solution, infinitely many solutions (a + 2)y + (a2-4)2 = (0-2) (b) If A = 4-1 0 a 2b a a be the augmented matrix of a linear system of equations then evaluate the values of a and b for which the linear system has no solution? exactly one solution? one parameter solution? two parameter...
(6) (Hand in!) Consider the linear function L : R3 → R3, given by the formula L(2)-Ax, where A is one of the following 3x 3 matrices: 0 -1 3 0 -21 0 1-2 3 1 -3 -1 2 0, 1 -1 1 21-3 Compute the determinant of each matrix. In each case what information does the determinant give you regarding: (i.) the invertibility of the function of L, (ii.) the volume of the parallelogram formed by passing the standard...
Consider the linear system of first order differential equations x' = Ax, where x= x(t), t > 0, and A has the eigenvalues and eigenvectors below. 4 2 11 = -2, V1 = 2 0 3 12 = -3, V2= 13 = -3, V3 = 1 7 2 i) Identify three solutions to the system, xi(t), xz(t), and x3(t). ii) Use a determinant to identify values of t, if any, where X1, X2, and x3 form a fundamental set of...
Section B - Answer any two questions. 2. (a) Consider the one-parameter family of nonlinear ordinary differential equations dr where a is a real parameter. i. Find all equilibrium points. ii. Sketch the bifurcation diagram, and indicate the behaviour of non- equilibrium solutions using appropriate arrows. ii. Find all bifurcation points and classify them. 10 Marks (b) Consider the second order differential equation i. Show that (1) can be written as the system of ordinary differential equations (y R for...
Consider the equation
1. Find the trace ? and the determinant ?. Sketch the graphs of
? vs. b and ? vs. b. Then sketch the graph of the curve ? = ? (b),
? = ?(b) in the ??-plane along with ?^2 = 4?, ? = 0 and ? = 0(? ?
0).
2. Find the intersection points of the the curve ? = ?(b),? =
?(b) with ?^2 = 4?, ? = 0 and ? = 0(? ?...
7. (14,5) A particular parametric curve is given by x=1-3, y = 21 +3 for -2 515 3. Sketch the curve using arrows to indicate the direction in which the curve is traced as increases. Then eliminate the parameter / to find a Cartesian equation of the curve. 8. (10,4) Find the equation of the plane through the point (-5, 4, 2) and with normal vector-31 +4j - k. Give your answer in both the vector equation of a plane...
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