How do I find the general solution of Y' = AY where the matrix A has real constant coefficients:• If no solution vectors are given: Look for all the eigenpairs of A.If Eigenvalues are real numbers andyou found a full set of eigenvectors (n Linearly Independent Eigenvectors for Anan: Note: If A is asymmetric matrix or if A has n different real eigenvalues you will always find a full set of eigenvectors),then proceed as follows:editvi. Write the general solution asWith each eigenpair (Ai, vi) build a solution of the form j;a linear conbination of the Ti. Rewrite the general solution in matrix form if needed. Solve IVP ifneeded.• If solution vectors are given: Make sure the given vectors are indeed solutions of the system. Build theSolution Matrix by placing the given vectors as columns. Find the Wronskian W (t) of your solutionmatrix by taking its determinant. Remember that you only need to evaluate this determinant at aparticular (convenient) point (by Abel's theorem). If the Wronskian is not zero, then your solutionmatrix is a Fundamental Matrix for your system and the columns of the Fundamental Matrix forma Fundamental Set (they are Lineraly Independent). Build the general solution in matrix form bymultiplying the F.M. by a constant vector č. Solve the IVP if needed.1. Consider the system4y1 – 2y2(1)1y1 + ly2Find a fundamental set of solutions. Write the general solution in matrix form and solve the IVP.

Answered: Image /qna-images/answer/0102b185-b69c-4304-95f3-b76e2cec6f66.jpg

Answered: 1. Consider the system Jui = 4y1 – 2y2…

1. Consider the system Jui = 4y1 – 2y2 (1 lyı + ly2 Find a fundamental set of solutions. Write the general solution in matrix form and solve the IVP.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

How do I find the general solution of Y' = AY where the matrix A has real constant coefficients:
• If no solution vectors are given: Look for all the eigenpairs of A.If Eigenvalues are real numbers and
you found a full set of eigenvectors (n Linearly Independent Eigenvectors for Anan: Note: If A is a
symmetric matrix or if A has n different real eigenvalues you will always find a full set of eigenvectors),
then proceed as follows:
editvi. Write the general solution as
With each eigenpair (Ai, vi) build a solution of the form j;
a linear conbination of the Ti. Rewrite the general solution in matrix form if needed. Solve IVP if
needed.
• If solution vectors are given: Make sure the given vectors are indeed solutions of the system. Build the
Solution Matrix by placing the given vectors as columns. Find the Wronskian W (t) of your solution
matrix by taking its determinant. Remember that you only need to evaluate this determinant at a
particular (convenient) point (by Abel's theorem). If the Wronskian is not zero, then your solution
matrix is a Fundamental Matrix for your system and the columns of the Fundamental Matrix form
a Fundamental Set (they are Lineraly Independent). Build the general solution in matrix form by
multiplying the F.M. by a constant vector č. Solve the IVP if needed.
1. Consider the system
4y1 – 2y2
(1)
1y1 + ly2
Find a fundamental set of solutions. Write the general solution in matrix form and solve the IVP.

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