I have the following ODE: $$ y'=y^2\cos t-\frac{1}{10}y $$ substituting $z=1/y$, I understand this is $z(t)=1/y(t)$. I know $t$ is the independent variable and $y$ is dependent variable.
What is the role of $z$ here? Does it change the dependent variable when $y(t)=1/z(t)$?
I saw in the book that it is written: $$ y'=\frac{-z'}{z^2} $$ does this mean that: $$ y'(t)=\frac{dz}{dy}\frac{dy}{dt} $$ is the relation $z(y(t))$?
Can someone explain the implicit differentiation in this expression? What are the dependent and independent variables here?
I understand how to finish this, but I want to know the intuition behind. I always confuse what are the dependent and independent variables when doing substitution, and to what variable I need to implicitly differentiate.
$y'$ in your equation means $ \frac {dy}{dt}$ since z is a func tion of the variable t
$$y'=\frac {dy}{dt}$$ $$y'=\frac {dy}{dz}\frac {dz}{dt}=\frac {dy}{dz}z'$$ Since you have $y=\frac 1 z$ then $$\frac {dy}{dz}=\frac {d}{dz}\frac 1z=-\frac {1}{z^2}$$ Therefore, $$y'=\frac {dy}{dz}z'=-\frac {z'}{z^2}$$ The simpliest way is to differentiate with respect to the variable t
$$y=\frac 1 z$$ Where z is a function of t $$\frac {dy}{dt}=\frac d {dt}\left(\frac 1 z\right)$$ $$y'=\frac {-z'}{z^2}$$