notes4 - y as a function of y . Thus, given the...

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MATH 1005A - Notes 4 Second-Order Equations Reducible to the First Order A second-order equation (linear or nonlinear) in which the dependent variable (e.g., y )does not appear explicitly can be reduced to a ¯rst-order equation for y 0 . Thus, given the second- order equation f ( x; y 0 ;y 00 )=0 ; let u ( x )= y 0 ( x ), then y 00 ( x )= u 0 ( x ), and the equation becomes f ( x; u; u 0 )=0 ,wh ichiso f the ¯rst order. For example, given y 0 y 00 + x =0 ,inwh i ch y does not appear, let u = y 0 to obtain the ¯rst-order equation uu 0 + x =0. Once u is determined by solving the ¯rst-order equation, y = Z y 0 ( x ) dx = Z u ( x
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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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