4-6. Homogeneous linear equations with constant coefficients

4-6. Homogeneous linear equations with constant...

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a Homogeneous linear equations with constant coefficients Definition . DE a n y ( n ) + a n 1 y ( n 1 ) + + a 2 y ' ' + a 1 y ' + a 0 y = 0 ( 1 ) is said to be n –order homogeneous linear differential equation with constant coefficients if the coefficients a i ,i = 1,2, …,n are real numbers and a n 0. It is known that the set of solutions of system (1) is n -dimensional vector space over R. Therefore to describe all its solutions it is enough to find n -linear independent solutions over R. Let y 1 ( x ) , y 2 ( x ) ,…, y n ( x ) be a system of solutions of the above DE over an interval I . If for constants c 1 ,c 2 ,…,c n the equality c 1 y 1 ( x ) + c 2 y 2 ( x ) + + c n y n ( x ) = 0 over I is true only for c 1 = c 2 = = c n = 0 then the system of functions y 1 ( x ) , y 2 ( x ) ,…, y n ( x ) is said to be linear independent over the interval I. Criterion. The system y 1 ( x ) , y 2 ( x ) ,…, y n ( x ) is linear independent over the interval I if and only if the following function (called the Wronskian determinant of this system) W ( x ) = det ( y 1 ( x ) y 2 ( x ) y 1 ' ( x ) y 2 ' ( x ) y 1 ( n 1 ) ( x ) y 2 ( n 1 ) ( x ) y n ( x ) y n ' ( x ) y n ( n 1 ) ( x ) ) . doesn’t vanish on I, that is W(x) is not zero whenever x in I.
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