Question

2 Find the general solution of the following linear systems. (1) X' = -a? where a is a non-zero constant. (2) X, 1 1 X®=(19)*

Differential equations- please show how to get eigenvectors after getting the eigenvalues and general solution

Answer 1

solution (1): Given system is, X' -al X C get et's for the the eigenvalues and eigenveeters matrix . 1-2 det (A->1) -a2 O 1-2 -> (2-15 +ahad I ai Now We will find the eigenvector for eigenvalue 2, =lt ai the

24 ut x = be the required eigenverfor 72 A X = 2 x =) (A-2, I) x = 0 ai a 11 => -ai is The augmented matrix l-ai - a² 04 | 0) ** ir tt -ai -ai -ai 1 I R2-R rail 0 to i'. The system is equivalent x, - aina FO c) 74 - aidz aina . 12 al choose H2 = 1, X .O If we the Then solution corresponding to this eigenvalue and eigen veetor is, X1 (t) = e (ltait lai

t => X,(t) é (los at tisin at > 0 t l-asinat tia lasat 17 e Gosat at isin at asinat aasat t 11 tiet e Abar Sinar of the linear system " The general solution then, - asinat alsat t x 1t) = ciet + C de sinat Goat and arbitrary Constants. where are G2 J solution : 2 Given system is, 0 x X 11 C. get et's for the the eigenvalues and eigenvertiss matrix det (A-nI) 0 (-12.0 * * * * , 1-2 0 = 0 => *]; 1-2

for 2,51 let X= (h () be the eigenver for r. AX= 2x, -> (A-2, I) X=0 -> 6.900 언 기 4=0 and 712 is arbitrary. choosing 22 - we get x = (i) -

We did not get two linearly independent the eigenvalue eigen rectors corresponding to ni the linearly independent system solve the To get 2nd eigenvector X2 we (A – 2, 1) X2 = x

o :)*)-(3) 언 and a2 is arbitrary → 24 = 1 we get choosing R2 FO 1 .. = X2 * a general solution is of the form The Xw). (**) mehr (9) *e*(!)] Note: The in the ot at X, te X2 x 11) general soll of the system repeated eigenvalue case is Genk x1 +62 (te ***, obtained by by solving the (A-XI) XL = x where X2 is system