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variation of parameters

variation of constants
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Mathematics Subject Classification

34A30 no label found34A05 no label found


I am not sure why this item lies in the "lacking a proof" list. Perhaps it is because of the title, which implies an ad-hoc method of solution. If one approaches this through matrix calculus, having discussed the solution of dZ/dt = MZ with a functional coefficient matrix M(t) and full matrix of solutions Z(t), the way to approach the inhomogeneous equation dW/dt = MW + F is to factor the solution matrix, writing W = ZQ.

Then dW/dt = dZ/dt Q + Z dQ/dt = MZQ + F. Supposing now that the homogeneous equation has already been solved, and therefore a known quantity, it remains to solve Z dQ/dt = F. A result from matrix calculus assures us that Z is invertible if it evolves from a basis of initial conditions, leaving the quadrature (that is, pure integration, no solving) Q = Q(0) + Integral(0,t)[Z^-1(s)F(s)ds].

Since the inverse of a matrix is its adjugate divided by its determinant (which in the case of an n'th order equation is a Wronskian), there results the nice formula in this posting.

Admittedly to convert the above into a proof, some details from matrix calculus have to be assumed, such as the existence of a solution in the form of the matrizant (Picard's method) or as a product integral (Euler's method). Likewise that d|Z|/dt = Trace(Z)|Z|, by having an exponential solution which never vanishes for analytic Z, guarantees the invertibility of Z.

- hvm

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