# It can be proved.docx - It can be proved the proof is...

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It can be proved (the proof is omitted) that thatfor the case of double root, this second equationmust be consistent, though the determinantdet(Aλ1I) = 0.Example:Given.We have der(.The system has adoubleeigenvaluesλ=λ1= 2,m1= 2. In this case, for the second solution weneed to solve the systems,Avˆ1,1 =λ1vˆ1,1,
()SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS2Avˆ1,0 =λ1vˆ1,0+vˆ1,1,becomesAvˆ1,1 = 2vˆ1,1,Avˆ1,0 = 2vˆ1,0 +vˆ1,1,We solvevˆ1,1= [c,d]Tfrom the first equation,.It is derived that rank{(A− 2I)} = 1 andvˆ1,1=r[2,1]T.The second equation forvˆ1,0= [a,b]Tbecomes.It is seen that this equation is consistent and hasthe solutions:
()SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS3vˆ,wherer,sare arbitrary constants. Letr= 2,s= 0,we getvˆ1,1= [4,2]T,vˆ1,0= [1,0]T.Note:For the case under discussion, one mayintroduce the matrixP= [vˆ1,1,vˆ1,0],we have.SincePis invertible, we may write.Setting as beforex=Pu
()SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS4withu,our system becomes
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Unformatted text preview: d u P = APu d t or d u − 1 AP u = ( λ 1 1 )u . = P d t λ 1 Hence, our system reduces to the uncoupled system d d ut 1 = λ 1u 1 + u 2 , dd u t 2 = λ 1u 2which has the general solution( ) ∗ SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS 5 u 2 = c 2e λ 1 t, u 1 = c 1e λ 1 t + c 2 t e λ 1t. Thus the general solution of the given system is x = P u = u 1 v ˆ1 ,1 + u 2 v ˆ1 , = ( c1 + c2 t )e λ1 t v ˆ1 ,1 +c 2eλ 1 t v ˆ1, . 2. Solutions for (n×n ) Homogeneous Linear System One may easily extend the ideas and results given in the previous sections to the general ( n × n ) system of linear equations: d x = A x , t (∈ I) (6.6) d t( ) ∗ SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS 6 where A is n × n matrix with real elements. Let us consider thesolution in the form: x( t ) = { x 1 (t ) , ··· ,x n (t )} = e t λv ....
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