L64 - MATLAB Examples Fall 2003 Math 308/501502 6...

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Fall 2003 Math 308/501–502 6 Higher-Order Linear Differential Eqs 6.4/4.6 Variation of Parameters Mon, 06/Oct c ± 2003, Art Belmonte Summary Let L [ y ] = y ( n ) ( x ) + n X j = 1 p j ( x ) y ( n - j ) ( x ) = g ( x ) be a nonhomogeneous n th-order linear differential equation in standard linear form (SLF). Recall that if y p is a particular solution of this nonhomogeneous equation and y h is a general solution to the associated homogeneous equation L [ y ] = 0, then a general solution of the nonhomogeneous equation is given by y = y p + y h . Variation of Parameters (VOP) The following method produces a general solution of the nonhomogeneous equation. First put the differential equation in standard linear form! 1. Obtain a fundamental set of solutions y f = [ y 1 ,..., y n ], a row vector, of L [ y ] = 0. Given a column vector of constants, c = [ c 1 ; ... ; c n ], form a general solution of this homogeneous equation, y h = y f c = n X k = 1 c k y k . 2. Form the column vector b = [0 ; 0 ; ; 0 ; g ], all of whose entries are zeros except the last one, which is the nonhomo- geneity g (right-hand side of the nonhomogeneous equation). Compute the Wronskian matrix M of y f .So lve Mv p = b for v p ( v prime), a column vector: vp = M \ b . 3. Integrate v p to obtain v , a column vector. (Don’t worry about constants of integration.) 4. A particular solution of the nonhomogeneous equation is y p = y f v . 5. A general solution of the nonhomogeneous equation is y = y p + y h . Use MATLAB for your computations! The operations involved in the variation or parameters procedure are tailor-made for MATLAB: matrix multiplication, solving linear systems, vector integration, etc. MATLAB Examples To facilitate computation of the Wronskian matrix, I wrote a function M-file named wron . Type “ help wron ” at a MATLAB prompt to learn about it. If you’d like to see the (short) code, type type wron ” at a MATLAB prompt.
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L64 - MATLAB Examples Fall 2003 Math 308/501502 6...

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