A first-order differential equation is separable when it can be rearranged so that all y-terms accompany dy and all x-terms accompany dx. Three common templates **Original form** | **Separated integral** | **Restriction or check** (dy)/(dx)=p(x)q(y) | (1)/(q(y)),dy= p(x),dx | Check q(y)=0 before dividing. (dy)/(dx)=(p(x))/(q(y)) | q(y),dy= p(x),dx | The original equation requires q(y) 0. (dy)/(dx)=(q(y))/(p(x)) | (1)/(q(y)),dy= (1)/(p(x)),dx | Require p(x) 0; check q(y)=0. If q(y_*)=0, then y=y_* may be a constant solution. Test it in the original differential equation before dividing by q(y). Method - Identify the interval and any points where the original equation is undefined. - Record possible constant solutions before dividing by a y-dependent factor. - Separate the variables and integrate both sides, including an arbitrary constant. - Apply an initial condition if one is given, then state the solution on a valid interval. An implicit relation between x and y is acceptable unless the question requires y explicitly as a function of x.

Separable Differential Equations

Junior College 1

A first-order differential equation is separable when it can be rearranged so that all \(y\)-terms accompany \(\mathrm{d}y\) and all \(x\)-terms accompany \(\mathrm{d}x\).

Three common templates

Original form	Separated integral	Restriction or check
\(\displaystyle\frac{\mathrm{d}y}{\mathrm{d}x}=p(x)q(y)\)	\(\displaystyle\int\frac{1}{q(y)}\,\mathrm{d}y=\int p(x)\,\mathrm{d}x\)	Check \(q(y)=0\) before dividing.
\(\displaystyle\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{p(x)}{q(y)}\)	\(\displaystyle\int q(y)\,\mathrm{d}y=\int p(x)\,\mathrm{d}x\)	The original equation requires \(q(y)\ne0\).
\(\displaystyle\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{q(y)}{p(x)}\)	\(\displaystyle\int\frac{1}{q(y)}\,\mathrm{d}y=\int\frac{1}{p(x)}\,\mathrm{d}x\)	Require \(p(x)\ne0\); check \(q(y)=0\).

Do not lose equilibrium solutions

If \(q(y_*)=0\), then \(y=y_*\) may be a constant solution. Test it in the original differential equation before dividing by \(q(y)\).

Method

Identify the interval and any points where the original equation is undefined.
Record possible constant solutions before dividing by a \(y\)-dependent factor.
Separate the variables and integrate both sides, including an arbitrary constant.
Apply an initial condition if one is given, then state the solution on a valid interval.

An implicit relation between \(x\) and \(y\) is acceptable unless the question requires \(y\) explicitly as a function of \(x\).

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