Please verfiy this differential substitution is mathamatically legal

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SUMMARY

The discussion centers on the legality of the substitution z = dy/dx in differential equations. The participants confirm that while one can express zdx = dy and integrate both sides to yield y = zx, this approach assumes z is constant. This assumption is incorrect in general, as z is dependent on x, which can lead to inaccuracies in the solution of the differential equation.

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  • Concept of dependent and independent variables
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John777
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In a differential equation in doing a substitution such as z=dy/dx can I do the following:zdx=dy

Integrate both sides

y=zx
 
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John777 said:
In a differential equation in doing a substitution such as z=dy/dx can I do the following:


zdx=dy

You can probably get away with doing this, yes.

John777 said:
Integrate both sides

Ah, but integrate both sides with respect to what? :wink:

John777 said:
y=zx

In taking this step, you have implicitly assumed that z is constant (i.e. that it does not depend on x). Since z = dy/dx, that's not going to be true in general, is it?
 

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