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Unformatted text preview: Inhomogeneous Linear Second Order Equations; Continued Method of Variation of Parameters; the Wronskian Determi- nant; Fundamental Sets Let w ( x ) and z ( x ) be two solutions of the homogeneous second order linear differential equation d 2 z dx 2 + p ( x ) dz dx + q ( x ) z = 0 . If we take a linear combination of these: c 1 w ( x ) + c 2 z ( x ) , and want to satisfy initial conditions y ( x ) = y , y ( x ) = y 1 at a point x = x , then we need c 1 w ( x ) + c 2 z ( x ) = y , c 1 w ( x ) + c 2 z ( x ) = y 1 . The condition for unique solvability, as always, is that the determinant det w ( x ) z ( x ) w ( x ) z ( x ) ! = w ( x ) z ( x )- w ( x ) z ( x ) 6 = 0 . The determinant expression W ( w, z, x ) = w ( x ) z ( x )- w ( x ) z ( x ) is called the Wronskian (determinant) of the solution pair w ( x ) , z ( x ). Proposition On any interval a < x < b where p ( x ) and q ( x ) are continuous, the Wronskian W ( w, z, x ), as a function of x , is either never zero or is identically zero. Proof We compute dW ( w,z,x ) dx = d dx ( w ( x ) z ( x )- w ( x ) z ( x )) = w ( x ) z ( x ) + w ( x ) z 00 ( x )- w 00 ( x ) z ( x )- w ( x ) z ( x ) 1 = w ( x ) (- p ( x ) z ( x )- q ( x ) z ( x ))- (- p ( x ) w ( x )- q ( x ) w ( x )) z ( x ) =- p ( x ) det w ( x ) z ( x ) w ( x ) z ( x ) ! + q ( x ) det w ( x ) z ( x ) w ( x ) z ( x ) ! =- p ( x ) W ( w, z, x ) + 0 =- p ( x ) W ( w, z, x ) , where, to obtain the last expression, we have used the differential equa- tion satisfied by both w ( x ) and z ( x ) and the fact that the determinant of a matrix with two identical rows is equal to zero. Thus, as a func- tion of x , the Wronskian W ( w, z, x ) satisfies a linear homogeneous first order differential equation d dx W ( w, z, x ) =- p ( x ) W ( w, z, x ) . Given any point x in the interval a < x < b , solution of this differ- ential equation gives W ( w, z, x ) = exp- Z x x p ( s ) ds W ( w, z, x ) , x ( a, b ) . Thus, if the Wronskian is zero at any point in ( a, b ), taking x to be that point, the formula shows that the Wronskian is zero at every other point x ( a, b ). On the other hand, if there is a point, call it x , where the Wronskian is not zero, then, since the exponential function is never zero, the formula shows that the Wronskian is non-zero at every other point, x , in ( a, b ). Thus the vanishing, or non-vanishing of the Wronskian W ( w, z, x ) associated with two solutions w ( x ) and z ( x ) of the homogeneous equa- tion d 2 z dx 2 + p ( x ) dz dx + q ( x ) z = 0 is a property of the two solution functions and not a property of the particular point x in question. Two solutions w ( x ) and z ( x ) for which the Wronskian is different from zero are said to constitute a fundamental (solution) set . When we have a fundamental solution set, w ( x ) and z ( x ), we can say without any fur- ther checking that y ( x, c, d ) = c w ( x ) + d z ( x ) is the general solution of the homogeneous equation in question....
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