Reducible Second Order ODE

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Reduction happens by considering one derivative to be another variable, thus reducing the number of derivates we're working with.

A general 2nd order differential is of the form $$F(x,y, y', y'') = 0$$but it can be converted into a 1st order differential under certain conditions.

Case 1 :

• if the independent term ($x$) is missing, then we can substitute $y^{\prime }=w$ which gives $y^{″}=w\frac{dw}{dy}$, so the equations becomes $$F\left( y, w, w \frac{dw}{dy} \right) = 0$$which is a 1st order and can be tackled

Case 2 :

• if the dependent variable, ($y$) is missing, then we can substitute $y^{\prime }=w$, and $y^{″}=w^{\prime }$, so we get $F(x,w,w^{\prime })=0$ which is a first order.

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