Solving Simple Differential Equation

Once a differential equation is set up, the next task is to actually solve it. This section covers the most basic first-order, first-degree equations — ones where the variables can be separated onto opposite sides and integrated directly.

Case 1 — the equation depends only on $x$. If it can be written as $\dfrac{dy}{dx} = F(x)$, rewrite it as $dy = F(x)\,dx$ and integrate both sides:

where $c$ is an arbitrary constant.

Case 2 — the equation depends only on $y$. If it can be written as $\dfrac{dy}{dx} = G(y)$, separate the variables and integrate with respect to $y$ to obtain a relation between $y$ and $x$ plus an arbitrary constant $c$.

Case 3 — variables separable. If the equation can be arranged into the form $g(y)\,dy = f(x)\,dx$ — every $y$-term (with $dy$) on one side and every $x$-term (with $dx$) on the other — integrate both sides independently: