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- Apr 14, 2013

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I need some help at the following exercise:

Let [tex]v_{1}[/tex], [tex]v_{2}[/tex] solutions of the differential equation [tex]y''+ay'+by=0[/tex] (where a and b real constants)so that [tex]\frac{v_{1}}{v_{2}}[/tex] is not constant.If [tex]y=f(x)[/tex] any solution of the differential equation ,use the identities of the Wronskian to show that there are constants [tex]d_{1}[/tex], [tex]d_{2}[/tex] so that:[tex]d_{1}v_{1}(0)+d_{2}v_{2}(0)=f(0)[/tex] (1), [tex]d_{1}v_{1}'(0) +d_{2}v_{2}'(0)=f'(0)[/tex] (2) and that each solution of the differential equation has the form:[tex]y=d_{1}v_{1}(x)+d_{2}v_{2}(x)[/tex].

Since [tex]\frac{v_{1}}{v_{2}}[/tex] is not constant, the Wronskian is not equal to zero, right? But how can I continue to show the relations (1) and (2)??