First-order

A first-order differential equation relates a function $y(x)$ to its derivative $y^{\prime }$. The key idea is separation of variables: when the equation can be written as $y^{\prime }=f(x)g(y)$, treat $dy/dx$ like a fraction, move all $y$ terms to one side and all $x$ terms to the other, then integrate both sides. For example, $dy/g(y)=f(x)dx$ becomes $\int dy/g(y)=\int f(x)dx$. After integrating, solve for $y$ and use any initial condition to find the constant. If $g(y)=0$ for some constant value, that value is also a solution, though it may not satisfy a nonzero initial condition.