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Unformatted text preview: y as a function of y . Thus, given the second-order equation f ( y; y ; y 00 ) = 0 ; let y ( x ) = u ( y ), then, by the chain rule, y 00 ( x ) = du dy dy dx = u du dy , and the equation becomes f ( y; u; uu ) = 0, or g ( y; u; u ) = 0, which is of the rst order. For example, given y 00 + ( y ) 2 y + 1 y = 0, in which x does not appear, let y ( x ) = u ( y ) to obtain the rst-order equation uu + u 2 y + 1 y = 0. Once u is determined by solving the rst-order equation, y is obtained by solving the rst-order equation y = u ( y )....
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- Winter '09