Mathematics

Problem

The differential equation

\displaystyle \frac{dy}{dx}=1+\frac{y}{x}

is not in a form where

x

and

y

can be isolated directly. Which substitution is most likely to lead to an equation in which

x

and a single new variable can be isolated on opposite sides? a)

v=y-x

v=\dfrac{y}{x}

v=xy

v=y+x^{2}

Intuitive explanation

This is a linear differential equation in disguise—the key is recognizing that the $\frac{y}{x}$ term suggests $y$ and $x$ are entangled in a ratio. The substitution $v = \frac{y}{x}$ (or equivalently $y = vx$ ) transforms a non-separable equation into one where $v$ and $x$ can be cleanly separated. Once you make this swap and differentiate using the product rule, the $y$ terms vanish and you're left with an equation in just $v$ and $x$ .

Intuitive explanation

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