find_the_fun
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In the context of "solutions by substitutions" in the examples there's a step I don't understand:
What is going on in these two examples? If y=ux then [math]\frac{dy}{dx}=u[/math] and [math]dy=u dx \neq udx + xdu[/math]
Same for the other, if [math]y=u^{-1}[/math] then [math]\frac{dy}{dx}=0[/math]?
If we let $$y = ux$$ then $$dy = u dx + x du$$
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we have $$u = y^{-1}$$ or $$y = u^{-1}$$. We then substitute $$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx} = -u^{-2} \frac{du}{dx}$$
What is going on in these two examples? If y=ux then [math]\frac{dy}{dx}=u[/math] and [math]dy=u dx \neq udx + xdu[/math]
Same for the other, if [math]y=u^{-1}[/math] then [math]\frac{dy}{dx}=0[/math]?