3.7we - 3.7 VARIATION OF PARAMETERS KIAM HEONG KWA This...

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KIAM HEONG KWA This section exhibits the method of variation of parameters to com- pute the general solution of a second-order linear nonhomogeneous equation from a given fundamental set of solutions of its associated homogeneous equation. It is more powerful than the method of unde- termined coefficients in the sense that the nonhomogeneous term is of a wider class of functions. Theorem 1 (Variation of Parameters) . Consider the equation (1) y 00 + p ( t ) y 0 + q ( t ) y = g ( t ) , where p ( t ) , q ( t ) , and g ( t ) are continuous functions on an open interval I . If y 1 ( t ) and y 2 ( t ) are linearly independent solutions of the associated homogeneous equation (2) y 00 + p ( t ) y 0 + q ( t ) y = 0 , then the general solution of (1) is given by (3) y ( t ) = - y 1 ( t ) Z y 2 ( t ) g ( t ) W ( y 1 ,y 2 )( t ) dt + y 2 ( t ) Z y 1 ( t ) g ( t ) W ( y 1 ,y 2 )( t ) dt. Proof. Instead of directly verifying that y ( t ) as given in (3) is the gen- eral solution of (1), we shall illustrate the method of variation of pa- rameters. To this end, recall that the general solution of (2) is given by y h ( t ) = c 1 y 1 ( t ) + c 2 y 2 ( t ) , where c 1 and c 2 are integration constants. Regarding the integration constants as parameters, we assume that (1) has a solution given by (4) y ( t ) = u 1 ( t ) y 1 ( t ) + u 2 ( t ) y 2 ( t ) , where u 1 ( t ) and u 2 ( t ) are differentiable functions to be determined. By imposing suitable hypothesis on u 0 1 ( t ) and u 0 2 ( t ), we shall determine these functions in terms of the known functions y 1 ( t ), y 2 ( t ), and g ( t ). To begin with, note that
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3.7we - 3.7 VARIATION OF PARAMETERS KIAM HEONG KWA This...

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