Diff Eqs 1 solutions - 2 1 dy = e x x dx 1c Rearrange this...

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Differential Equations 1 – Selection of Solutions 1c. 2 1 x e dx dy x - = Rearrange this equation: 2 x xe dx dy - = This can now be integrated directly. Integrate both sides with respect to x : dx xe y x - = 2 Make a substitution for the awkward part: let 2 x u - = dx xe y u = du du dx xe u = du x xe u - = 2 1 since x dx du 2 - = and therefore x du dx 2 1 - = du e u - = 2 1 c e u + - = 2 1 c e x + - = - 2 2 1 where c is an arbitrary constant Alternatively , the R.H.S. can be integrated in one step, but it may not be obvious. 2 x xe dx dy - = c e y x + - = - 2 2 1 where c is an arbitrary constant How do we make that last step? Recognise that in the expression dx xe x - 2 , the x in front of the 2 x e - looks like the derivative of the index, ( 29 2 x - . Instead of integrating, it might be easier to think of this in reverse – ie, what, when differentiated, gives 2 x xe - ? Differentiating x e gives x e
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Differentiating x e 2 gives x e 2 2 Differentiating 2 x e gives 2 2 x xe Differentiating 2 x e - gives 2 2 x xe - - Think about the connection between differentiating and integrating.
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