Upload your study docs or become a

Course Hero member to access this document

**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 ....

View
Full Document