separable_equations

Separable First-Order Equations

Definition

A first-order ODE is separable if it can be written in the form

$$g(y)\frac{dy}{dx}=f(x),y(x_0)=y_0,$$

where $g(y)$ is independent of $x$ and $f(x)$ is independent of $y$.

Intuition

Separability means the $x$-dependence and $y$-dependence can be pulled to opposite sides of the equation. The differential $dy/dx$ is treated as a fraction, and each “side” is integrated independently — turning a differential equation into two ordinary integrals.

Formal Description

Integration from $x_0$ to $x$ and the substitution $u=y(x)$ yield

$${\int }_{y_0}^{y}g(u)du={\int }_{x_0}^{x}f(x)dx.$$

This can often be solved analytically for $y=y(x)$.

A practical shorthand: treat $\frac{dy}{dx}$ as a fraction to obtain the separated form

$$g(y)dy=f(x)dx,$$

which is then integrated directly on each side.

Key Results

• Separability is a sufficient condition for an analytic solution by direct integration.
• The resulting algebraic equation may or may not be solvable explicitly for $y$.

Applications

• Exponential growth and decay: $\dot{y}=ky$.
• Logistic population growth: $\dot{y}=ky(1-y/N)$.
• Deriving the integrating factor $\mu$, which itself satisfies $d\mu /dx=p(x)\mu$.

Trade-offs

• Only applicable when the equation can be separated; many physically relevant ODEs are not separable.
• Even when separated, the resulting integrals may lack closed forms.
• The implicit relation obtained after integration may be difficult or impossible to solve explicitly for $y(x)$.

Links

• Linear First-Order ODE