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Unformatted text preview: Math 334 Lecture #17 3.6,4.4: Variation of Parameters Theorem. If y 1 , y 2 , . . . , y n are linearly independent solutions of L [ y ] = y ( n ) + p 1 ( t ) y ( n- 1) + + p n- 1 ( t ) y + p n ( t ) y = 0 , then a particular solution of L [ y ] = g ( t ) is Y p = n X m =1 y m ( t ) Z g ( t ) W m ( t ) W ( t ) dt where W ( t ) = W ( y 1 , . . . , y n )( t ) is the Wronskian, and W m ( t ) is the determinant of the matrix for W ( t ) with the m th column replaced by [0 , , . . . , , 1] T . Remark. For second-order equations, the variation of parameters formula for a particular solution is Y p =- 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. [NOTES: By Abels Theorem and the linear independence of y 1 and y 2 , the Wronskian W ( t ) is never zero on I , so that the integrals make sense. Also, the form of Y p says that solving the nonhomogeneous ODE L [ y ] = g ( t ) is no harder than solving the homogeneous ODE L [ y ] = 0.] Proof for Second-Order Equations. Set Y p = uy 1 + vy 2 . Substitution of this guess for a particular solution into L [ y ] = g ( t ) gives only one differential equation in the two functions u and v ....
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