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Unformatted text preview: MA366 Sathaye Notes on Chapter 4,7 Brief Notes on 4.1, 4.2 1. We have already studied second order linear equations in great detail. Fortunately, our methods naturally extend to higher order equations without much modification. We give a brief summary of this. 2. Existence and Uniqueness theorems. Suppose we have an n-th order linear equation ( D n + a 1 ( t ) D n- 1 + + a n ( t )) y = g ( t ) where n 1. Assume that The functions a 1 ( t ) , ,a n ( t ) ,g ( t ) are continuous on some interval I containing t and we are given initial conditions y ( t ) = y ,Dy ( t ) = y (1) , ,D n- 1 y ( t ) = y ( n- 1) . Then there is a unique solution y = ( t ) valid on I and satisfying the initial conditions. 3. If we have a homogeneous linear equation of order n ( D n + a 1 ( t ) D n- 1 + + a n ( t )) y = 0 where n 1. and if the functions a 1 ( t ) , ,a n ( t ) are continuous on some interval I , then there is a set of n independent functions y 1 ( t ) , ,y n ( t ) which are solutions to the equation on I . Moreover, the independence of these functions can be checked by checking the value of their Wronskian W ( t ) = W ( y 1 ( t ) , ,y n ( t )) at any point of I to be non zero. The Wronskian W ( t ) is either nowhere zero in I or identically zero in I . In the latter case, the functions are linearly dependent on I . The Wronskian can be calculated up to a constant multiplier c without finding the solutions by using generalized Abels theorem which gives the formula: W ( y 1 ,y 2 , ,y n ) = ce- R a 1 ( t ) dt ....
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