1. The problem statement, all variables and given/known data
A = the matrix {{1 , -1},{4,-3}} and x' = Ax
2. Relevant equations
3. The attempt at a solution
The eigenvalue of A is -1, and one linearly independent solution is x1(t)= e^(-t) * [1 2]^T
To find the second solution, the book says to try x2(t) = te^(-t)*u1 + e^(-t)*u2. I substituted x2 into x2' = A x2 and got Ate^(-t)u1 = (-te^(-t) + e^(-t))u1 and Ae^(-t)u2 = -e^(-t)u2 after equating like terms. From there, I'm supposed to derive the relations (A+I)u1 = 0 and (A+ I)u2 = u1. How am I supposed to go from Ate^(-t)u1 = (-te^(-t) + e^(-t))u1 to (A+ I)u1 = 0. I got (A+I)u2 = 0 since it looks like u2 is an eigenvector for A's eigenvalue of -1. I'm so lost right now.
If it helps, this problem is in section 9.5 problem 35 of Fundamentals of Differential equations and Boundary value problems
A = the matrix {{1 , -1},{4,-3}} and x' = Ax
2. Relevant equations
3. The attempt at a solution
The eigenvalue of A is -1, and one linearly independent solution is x1(t)= e^(-t) * [1 2]^T
To find the second solution, the book says to try x2(t) = te^(-t)*u1 + e^(-t)*u2. I substituted x2 into x2' = A x2 and got Ate^(-t)u1 = (-te^(-t) + e^(-t))u1 and Ae^(-t)u2 = -e^(-t)u2 after equating like terms. From there, I'm supposed to derive the relations (A+I)u1 = 0 and (A+ I)u2 = u1. How am I supposed to go from Ate^(-t)u1 = (-te^(-t) + e^(-t))u1 to (A+ I)u1 = 0. I got (A+I)u2 = 0 since it looks like u2 is an eigenvector for A's eigenvalue of -1. I'm so lost right now.
If it helps, this problem is in section 9.5 problem 35 of Fundamentals of Differential equations and Boundary value problems
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