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**Unformatted text preview: **1 Lecture 15 1.1 Linear independence and Wronskian Recall that the determinant of a 2 & 2 matrix is de&ned by det & a 11 a 12 a 21 a 22 ¡ = ¢ ¢ ¢ ¢ a 11 a 12 a 21 a 22 ¢ ¢ ¢ ¢ = a 11 a 22 ¡ a 12 a 21 : Consider a system of two linear equations in two unknowns: a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 Recall that the foregoing system has a unique solution for every selection of b 1 and b 2 if and only if ¢ ¢ ¢ ¢ a 11 a 12 a 21 a 22 ¢ ¢ ¢ ¢ 6 = 0 : Consider a pair of continuously di/erentiable functions f ( x ) and g ( x ) ; a < x < b . Suppose that f ( x ) and g ( x ) are linearly dependent for a < x < b . Then there are constants & and ¡ such that & 2 + ¡ 2 6 = 0 and &f ( x ) + ¡g ( x ) = 0 ; for every a < x < b: Di/erentiate both sides of the foregoing equation; we get: &f ( x ) + ¡g ( x ) = 0 ; for every a < x < b: So, &f ( x ) + ¡g ( x ) = 0 &f ( x ) + ¡g ( x ) = 0 : Hence, we have a system of two equations in two unknowns & and ¡ , both not zero. It is well-known that a homogeneous n & n system of linear equations¡ in our case n = 2 ¡ has a non-trivial solution if and only if its determinant is zero: ¢ ¢ ¢ ¢...

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