# Variation of Parameters for System of 1st order ODE

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Kreyszig Advanced Engineering Mathematics shows the variation of parameter method for a system of first order ODE: $$\underline{y}' = \underline{A}(x)\underline{y} + \underline{g}(x)$$ The particular solution is: $$\underline{y}_p = \underline{Y}(x)\underline{u}(x)$$ where $\underline{Y}(x)$ is the homogeneous solution and: $$\underline{u}(x) = \int_0^x \underline{Y}^{-1}\underline{g}$$ But, for for a single 2nd order non-homogeneous ODE $y'' + p(x) y' + q(x) y = r(x)$ the solution is:$$y_p = y_1 \int \frac{y_2}{W}r(x)dx + y_2 \int \frac{y_1}{W}r(x)dx$$ where $y_1, \, y_2$ are the solutions to the homogeneous equation and $W$ is the Wronskian of the homogeneous solutions.

Why the definite integral in the case of systems but indefinite integral in the case of a single 2nd order ODE? The book offers no explanation.

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## Answers and Replies

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I checked my undergrad math book and it uses indefinite integral for systems. To (hopeful) gain some insight, I worked a couple of problems (one from Kreyszig and one I converted to a system from a single 2nd order ODE) and in both cases, the lower limit of integration produced extraneous solutions that could be absorbed in the homogeneous solution, which suggests the definite integral is worthless. But, through some web searching I found an example using definite integral for initial value problems. Thus, it appears the definite integral is useful if the stated problem has initial conditions. I wonder if the same is true for variation of parameters for non-systems?