One shape of equation can be solved by nothing more than integration: the shape in which the two variables can be pushed to opposite sides. If the right-hand side is a function of the input times a function of the unknown, then everything involving the unknown can be gathered with the derivative and everything involving the input left behind. Recognising the shape is the whole skill; after that the work is two integrals.

A pile of mixed coins can be counted quickly if it can first be sorted into denominations. Separation is that sorting step, and the equations it works on are exactly the ones whose right-hand side comes apart into a product with one factor of each kind.

The field of $y'=\tfrac35xy$, whose right-hand side is $x$ times $y$. That product is what makes it separable, and the solutions are the curves $y=Ce^{3x^{2}/10}$, which open away from the axis, together with the constant zero.

A first-order equation is separable when it can be written as a function of the input multiplied by a function of the unknown. Dividing by $h(y)$ and multiplying by $dx$ gathers each variable on its own side, ready to be integrated. Whether an equation is separable is a question about algebra, not about difficulty: many separable equations have integrals nobody can do in closed form.

• $y'=xy$ is separable with $g(x)=x$ and $h(y)=y$. So is , with .

Separable means the right-hand side is a function of the input times a function of the unknown. A sum is not a product, and no amount of rearranging turns one into the other.

Some equations separate only after the right-hand side is rewritten. Exponentials of sums and common factors are the usual hiding places.

Great work! You now know how to:

• recognise a separable equation by finding the product hiding in it
• name the two factors and write the separated form
• tell a sum, which does not separate, from a product that does
• say why separable does not mean solvable in closed form