Solve separable first-order differential equations and apply an initial condition to find the particular solution

What this dot point is asking

SCSA wants you to recognise a separable equation, carry out the separation and integration carefully, and apply a given condition to find the constant.

Recognising separability

An equation is separable when the right side factors into a part depending only on $x$ times a part depending only on $y$. Then all the $y$-dependence can be moved to one side with $dy$ and all the $x$-dependence to the other with $dx$.

Treating $\tfrac{dy}{dx}$ as a ratio of differentials is a valid shortcut here; the underlying justification is the chain rule.

Integrating both sides

Integrate each side with respect to its own variable. A single arbitrary constant suffices (combine the two constants of integration into one). The result is often an implicit relation between $x$ and $y$; rearrange for $y$ explicitly only if the algebra permits.

Applying an initial condition

Watching for lost solutions

Dividing by $h(y)$ assumes $h(y) \neq 0$. Any value of $y$ making $h(y) = 0$ is a constant equilibrium solution that the separation step can hide; note it separately.