linear_first_order

Linear First-Order ODE

Definition

A linear first-order equation with initial condition in standard form:

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

Intuition

Multiplying by the integrating factor “balances” the left-hand side so it collapses into a single derivative $\frac{d}{dx}[\mu y]$, after which the equation can be integrated directly. Geometrically, the factor $\mu$ warps the solution space so that the ODE becomes a trivially solvable exact derivative.

Formal Description

All linear first-order ODEs can be integrated using an integrating factor $\mu =\mu (x)$. Multiplying the equation by $\mu (x)$ and requiring

$$\mu (x)\left[\frac{dy}{dx}+p(x)y\right]=\frac{d}{dx}\left[\mu (x)y\right]$$

yields a directly integrable equation. The integrating factor is

$$\mu (x)=e^{{\int }_{x_0}^{x}p(x)dx},$$

and the solution is

$$y(x)=\frac{1}{\mu (x)}\left(y_0+{\int }_{x_0}^x\mu (x)g(x)dx\right).$$

Key Results

• Any linear first-order ODE can be solved by the integrating factor method.
• The integrating factor satisfies $\frac{d\mu }{dx}=p(x)\mu$, which is itself a separable equation.
• The method applies even when the equation is not separable.

Applications

• Mixing and decay problems (e.g.\ concentration in a tank, radioactive decay with external input).
• RC circuit analysis: $\dot{q}+q/(RC)=V(t)/R$.
• Population models with immigration or harvesting terms.

Trade-offs

• Requires $p(x)$ and $g(x)$ to be continuous on the interval of interest.
• The integrals $\int pdx$ and $\int \mu gdx$ may not have closed forms, requiring numerical integration.
• Only applies to linear equations; nonlinear first-order ODEs need separate methods (e.g.\ separation of variables).

Links

• Separable First-Order Equations
• Inhomogeneous Second-Order ODE