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Unformatted text preview: Problem 7.4.6. Consider the vector functions x (1) ( t ) = parenleftbigg t 1 parenrightbigg and x (2) ( t ) = parenleftbigg t 2 2 t parenrightbigg . The Wronskian of these functions is W [ x (1) , x (2) ]( t ) = det parenleftbigg t t 2 1 2 t parenrightbigg = t (2 t )- 1( t 2 ) = t 2 . Since W [ x (1) , x (2) ]( t ) = t 2 negationslash = 0 precisely when t negationslash = 0 , the vectors x (1) ( t ) and x (2) ( t ) , i.e., the vec- tor functions x (1) and x (2) evaluated at t , are linearly independent precisely when t negationslash = 0 . Hence the vector functions x (1) and x (2) are linearly independent in any interval because any interval contains at least a nonzero t-value. Suppose x prime = P ( t ) x or, more explicitly, parenleftbigg x prime 1 ( t ) x prime 2 ( t ) parenrightbigg = parenleftbigg p 11 ( t ) p 12 ( t ) p 21 ( t ) p 22 ( t ) parenrightbiggparenleftbigg x 1 ( t ) x 2 ( t ) parenrightbigg is the system of homogeneous differential equations satisfied by x (1) and x (2) . Then by Abels....
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