4-5. Reduction of order - Reduction of order 1 In case...

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Reduction of order 1 °. In case equation on form у ( n ) = f ( x ) solved by n times integrating. 2°. In case the dependent variables у is missing from our differential equation, that is x, у ' , у ) =0 F ¿ , we make the substitution у' = р . This entails у = p ' . The differential equation is reduced to first order. Example . Solve the differential equation x y ' ' y ' = 3 x 2 using reduction of order. Solution. Notice that the dependent variables у is missing from the differential equation. We set у' = р ( x ) and у = p ' , so that the equation becomes xp' p = 3 x 2 . New equation is first order linear. We write it in standard form p' 1 x p = 3 x . Solving this equation we obtain p ( x ) = 3 x 2 + Cx . Now we recall у' = р , so we make that sub substation. Then y ' = 3 x 2 + Cx. Hence y = x 3 + C 2 x 2 + D = x 3 + E x 2 + D 3°. In case the variable x is missing from our differential equation, that is F(y, у', у") = 0, we make the substitution у' = р ( y ) . (Pay attention, here y’ is considered as a function of y , not of x )
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This time the corresponding substitution for у is a bit different. To wit
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