MATH201 Chapter 3: ch3-p1.pdf
16 Apr 2014
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A linear, second order di erential equation can be written in the form a2(x) d2y dx2 + a1(x) dy dx. + q(x) y = g(x). (3. 1) d2y dx2 + p (x) dy dx. The function g(x) is sometimes called a forcing function. Recall that a linear equation is called homoge- neous if the forcing term g(x) 0, and nonhomogeneous otherwise. It is sometimes convenient to write (3. 1) in the form where l is the di erential operator de ned by: L[y] = g, d2 dx2 + p (x) d dx. Di erential operators act on functions to give other functions. Let d(l) denote the domain of l. then, for. An operator l, di erential or otherwise, is called linear if it satis es: for all 1, 2 d(l): l[ 1 + 2] = l[ 1] + l[ 2], l[c ] = cl[ ], for all d(l), c r.
