In this example we will consider the transformation of a second-order ODE to a system of two first-order ODEs. The second-order differential equation is given by

This system of first-order differential equations is integrated to solve the second-order ODE Equation (6.34), using any of the methods described in the previous chapters.

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Let and d²y a(x)- +b(x). = c(x). dx² dy dx y(x) = y₁(x) dyi dx Subsequently, the second-order equation is transformed to the system dyi dx dy2 dx = y₂(x). = y₂(x), = c(x) - b(x)y₂(x) a(x) (6.34) (6.35) (6.36) (6.37)