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Unformatted text preview: Laplace Solution of Initial Value Problems We will be concerned here with linear n-th order differential equations with constant coefficients. Using differential operator notation, these take the form p ( D ) y ( x ) = g ( x ) , where p ( D ) y is defined by p ( D ) y ≡ a d n y dx n + a 1 d n- 1 y dx n- 1 + ··· + a n- 1 dy dx + a n y. The symbol D represents the operation of differentiation: d dx . Using the symbol D m to denote the m-th derivative, D m = d m dx m , we can write p ( D ) ≡ a D n + a 1 D n- 1 + ··· + a n- 1 D + a n I , a polynomial in the symbolic operator D ( I = D is the identity operator) with the same coefficients as the standard character- istic polynomial p ( r ) = a r n + a 1 r n- 1 + ··· + a n- 1 r + a n . The coefficients a k , k = , 1 , ··· , n are constants, ordinar- ily real. Such a linear n-th order differential equation is said to be homogeneous if g ( x ) ≡ 0 and inhomogeneous otherwise. The homogeneous equation has general solutions which take the 1 form (here c is an n-dimensional vector whose components are arbitrary constants c k , k = 1 , 2 , ··· , n ) y ( x, c ) = c 1 y 1 ( x ) + c 2 y 2 ( x ) + ··· + c n y n ( x ) , the y k ( x ) , k = 1 , 2 , ··· , n , being linearly independent solutions of that equation. If y p ( x ) is a particular (i.e., any ) solution of the inhomogeneous equation then its general solution takes the form y ( x, c ) + y p ( x ). An initial value problem (for either the homogeneous or the inhomogeneous equation) is specified by singling out a particular value, say x , of the independent variable x and specifying that the solution y ( x ) should satisfy n conditions y ( k ) ( x ) ≡ d k y dx k ( x ) = y k , k = 1 , 2 , ··· , n, the y k being given (ordinarily real) numbers. If the formula for a general solution is substituted into this list of initial conditions there results a set of n linear algebraic equations in the coeffi- cients c k which, as a consequence of the linear independence of the solutions y k ( x ), always has a unique solution. In this way a unique solution of the initial value problem is obtained.unique solution of the initial value problem is obtained....
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This note was uploaded on 01/23/2012 for the course MATH 4254 taught by Professor Robinson during the Spring '10 term at Virginia Tech.
- Spring '10