Similarity transformation-proof of equivalence

Question

I am getting stuck with following problem:

Show that \begin{align} \dot{x} = f(x/t) \end{align} is equivalent to \begin{align} \dot{y} = (f(y) − y)/t \end{align} using the transformation \begin{align} y := x/t. \end{align}

Any help/hint is welcomed.

Answer 1

If $y=\frac{x}{t}$, we have $$\begin{align} \dot{y} &= \frac{\dot{x}t-x}{t^{2}} \tag{derivative of a quotient}\\ &= \frac{f\left(\frac{x}{t}\right)t-x}{t^{2}} \tag{$\dot{x}=f(x/t)$}\\ &= \frac{f\left(\frac{x}{t}\right)t}{t^{2}}-\frac{x}{t}\cdot\frac{1}{t}\\ &= \frac{f(y)}{t} - y\cdot\frac{1}{t} \tag{since $t\neq 0$}\\ &= \frac{f(y)-y}{t} \end{align}$$

Answer 2

sometimes a small transformation of a problem makes it easier to approach. here you may write the transformation as $yt=x$. then, differentiating wrt t we get (using the product rule) $$ y+t\dot{y}=\dot{x}=f(y) $$ and the required answer is obtained by a simple algebraic manipulation