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

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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