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fiksx

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## Homework Statement

$$y'=-\frac{1}{10}y+(cos t)y^2$$

when doing substitute for ##z=\frac{1}{y}##

I understand this is ##z(t)=\frac{1}{y(t)}##

I know t is independent variable and y is dependent variable

but I want to know what is z role here, is it change the dependent variable?

when ##y(t)=\frac{1}{z(t)}##

then

I saw in book it is written ##y'=\frac{-z'}{z^2}##, is this means ##y(t)'=\frac{dz}{dy}\frac{dy}{dt}##?

is the relation is ##z(y(t))##? can someone explain the implicit differentiation in this expression? and what is dependent and independent variable here? thanks!

I understand how to finish this. but I want to know the intuition behind this. I always confuse when doing substitution, what is the dependent and independent variable, and to what variable I need to implicitly differentiate.

## Homework Equations

## The Attempt at a Solution

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