System of differential equations

Oct 2012
257
19
israel
Let x'=Ax be a system of differential equations, where A is a real constant 3x3 matrix with exactly two different real eigenvalues.

which one of the following statements must hold:(only one of them follows from the data)

a)there is no base of solutions in the form V(e^at) where a is a real number and V is a real vector

b)there is a solution in the form V(e^at) where a is a real number and V is a real vector

I think that the two follows

Thank's in advance
 
Dec 2013
2,000
757
Colombia
I would suggest a). Because two distinct real eigenvalues implies that one of them is repeated (because it's a 3x3 matrix, not a 2x2) and so I expect a solution of the form
$$\vec X = c_1\vec{V_1}e^{a_1t} + c_2\vec{V_2}e^{a_2t} + c_3t\vec{V_3}e^{a_2t}$$
Mind you, if $c_2 = c_3 = 0$ then you have the given form I suppose.
 
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Oct 2012
257
19
israel
Yes but the first statement is also true
 
Dec 2013
2,000
757
Colombia
Yes, there is no basis of all solutions with only one exponential term. But is there any reason why $$\vec X = c_1\vec{V_1}e^{a_1t}$$ wouldn't be a solution for every value of $c_1$? In this case, the given expression is a basis for a subset of all possible solutions.
 
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