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.
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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Transcribed Image Text: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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