A First-Order ODE Solver helps students understand how differential equations describe rates of change. Geometrically, the equation $y’ = f(x, y)$ assigns a specific slope to every point $(x, y)$ in the plane. The Slope Field (or Direction Field) visualizes these slopes with short line segments. A particular solution is a curve that follows these “flow lines” tangent to the slopes at every point.

1. Separable Equations

A differential equation is Separable if it can be factored into a function of solely $x$ and a function of solely $y$: $$ \frac{dy}{dx} = g(x)h(y) \implies \frac{1}{h(y)} dy = g(x) dx $$ We solve these by integrating both sides independently. This is often the first technique taught in undergraduate calculus.

2. Linear Equations

A Linear First-Order ODE fits the standard form: $$ \frac{dy}{dx} + P(x)y = Q(x) $$ To solve this, we calculate an Integrating Factor, $\mu(x) = e^{\int P(x)dx}$. Multiplying the entire equation by $\mu(x)$ allows us to collapse the left side into the derivative of a product, making it integrable.

3. Exact Equations

An equation written as $M(x, y)dx + N(x, y)dy = 0$ is Exact if the partial derivatives satisfy the condition: $$ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} $$ If exact, there exists a potential function $\Psi(x, y) = C$ whose total differential creates the ODE. We find $\Psi$ by integrating $M$ with respect to $x$ and $N$ with respect to $y$.