First-Order ODE Methods: Complete Review and Guide

A first-order ordinary differential equation involves only the first derivative dy/dx, and the classic methods for solving them are separation of variables, integrating factors for linear equations, and exact equations. Each method recognizes a structural pattern in the equation and exploits it to reduce the problem to integration.

These methods matter because first-order models describe how quantities change: population growth, cooling objects, draining tanks, and electric circuits all lead to first-order ODEs. Recognizing which method applies is the fundamental skill of the subject.

Key Concepts

• Separable equation. dy/dx = g(x)h(y): all x-terms can be moved to one side and all y-terms to the other, giving ∫ dy/h(y) = ∫ g(x) dx.
• Linear first-order equation. dy/dx + P(x)y = Q(x). The standard form must have coefficient 1 on dy/dx before applying the method.
• Integrating factor. μ(x) = e^{∫P(x)dx}. Multiplying the equation by μ makes the left side the exact derivative d/dx(μy).
• Exact equation. M(x, y)dx + N(x, y)dy = 0 with ∂M/∂y = ∂N/∂x, so the left side is dF for some potential function F.
• General solution. Always contains one arbitrary constant C, because one integration was performed.

Formulas

• Integrating factor: μ(x) = e^{∫P(x)dx}, then d/dx(μy) = μQ, so y = (1/μ)(∫μQ dx + C).
• Separable: ∫ dy/h(y) = ∫ g(x) dx + C, valid where h(y) ≠ 0; check h(y) = 0 separately for constant solutions.
• Exactness test: M dx + N dy = 0 is exact iff ∂M/∂y = ∂N/∂x on a simply connected domain.
• Assumptions and limits: the integrating factor formula requires the standard form dy/dx + P(x)y = Q(x); dividing by the coefficient of dy/dx first is mandatory.

Worked Example

Solve dy/dx + 2y = e^x by the integrating factor method.

1. Confirm standard form: dy/dx + P(x)y = Q(x) with P(x) = 2 and Q(x) = e^x. The coefficient of dy/dx is already 1.
2. Integrating factor: μ = e^{∫2dx} = e^{2x}.
3. Multiply through: e^{2x} dy/dx + 2e^{2x} y = e^{3x}. The left side is d/dx(y e^{2x}).
4. Integrate: y e^{2x} = ∫e^{3x} dx + C = e^{3x}/3 + C.
5. Solve for y: y = e^x/3 + C e^{−2x}.
6. Verify: dy/dx = e^x/3 − 2C e^{−2x}, so dy/dx + 2y = e^x/3 − 2C e^{−2x} + 2e^x/3 + 2C e^{−2x} = e^x. Correct.

Common Misconceptions

• Forgetting the +C. Every first-order method ends with an indefinite integral, so the constant of integration is part of the answer, not optional decoration.
• Applying the integrating factor without standard form. For 2dy/dx + 4y = e^x you must first divide by 2; using P = 4 directly gives a wrong μ.
• Dividing by h(y) blindly in separable equations. Zeros of h(y) give constant solutions that the separated integral can miss; check them separately.

Real-World Applications

• Newton's cooling: dT/dt = −k(T − T_env) is linear and solved by an integrating factor, giving exponential approach to room temperature.
• RC circuits: the capacitor voltage satisfies a linear first-order ODE with the same exponential solution shape.
• Population models: dP/dt = kP separates immediately to give exponential growth P = P0 e^{kt}.

First-order ODEs yield to pattern recognition: separate when the variables split, use μ = e^{∫Pdx} for linear equations in standard form, and test ∂M/∂y = ∂N/∂x for exactness. One integration, one constant C, and always verify by substituting back.