Substitutions

A substitution is simply a change of variable: you trade the unknown, or some combination of it, for a new letter so the equation turns into one you already know how to solve. It's the same move as $u$-substitution in an integral, applied to a whole equation, and it's how you rescue equations that aren't linear or separable as written. All we're really saying is: if $y$ and $t$ keep showing up in the same combination, give that combination its own name and solve for it instead. The formula $y = vt \implies y' = v + t\,v'$ looks fussy, but it's just two pieces: $v$ (the ratio $\frac yt$, our new unknown) and the product rule applied to $y = vt$ (which is where $v + t\,v'$ comes from). After that switch, the equation usually separates. It looks complex, but it's just a renaming and one derivative, so bear with it.

Homogeneous equations (in this sense, which has nothing to do with the "homogeneous" from linear equations): if $y'$ depends only on the ratio $\frac yt$, set $v = \frac yt$. Then $v + tv' = F(v)$, which separates:

Think about why that works. The right side only ever sees $\frac yt$, so after the switch it only sees $v$. The left side picks up a $t$ from the product rule, and that lonely $t$ is all the $t$-dependence there is. So the $v$'s and $t$'s land on opposite sides.

A common mistake. It's tempting to replace $y'$ with $v'$, because $v$ replaced $y$. It isn't that simple: $y = vt$, so the product rule gives $y' = v + tv'$. Drop the extra $v$ and you'll be solving the wrong equation.

Linear combinations: if $y' = G(at + by)$, set $u = at + by$. Then $u' = a + by' = a + bG(u)$, which is separable.

Bernoulli's substitution in the last lesson is another example of the same idea. Spotting the right substitution is a skill that grows with practice.

Scaling substitutions like $v = y/t$ are how physicists find similarity solutions: patterns that look the same at every scale, such as the spread of heat from a point.