First-Order Ordinary Differential Equations

What is an ODE?

A differential equation involves a function and its derivatives. It models systems that change continuously.

Order = highest derivative present

General first-order ODE: dy/dx = f(x, y)

Separable Equations

If the ODE can be written as g(y)dy = f(x)dx, integrate both sides.

Example: dy/dx = 2xy

dy/y = 2x dx
∫dy/y = ∫2x dx
ln|y| = x² + C
y = Ae^(x²)    (A = e^C)

Linear First-Order ODEs

Standard form: dy/dx + P(x)y = Q(x)

Solution using integrating factor μ = e^(∫P dx):

1. Compute μ(x) = e^(∫P(x)dx)
2. Multiply both sides by μ
3. Left side becomes d/dx[μy]
4. Integrate both sides
5. Solve for y

Example: dy/dx + 2y = 4

• P = 2, μ = e^(2x)
• d/dx[e^(2x)y] = 4e^(2x)
• e^(2x)y = 2e^(2x) + C
• y = 2 + Ce^(-2x)

Initial Value Problems (IVP)

Use initial condition y(x₀) = y₀ to find the constant C.

Engineering Applications

Exponential Decay/Growth

dN/dt = kN → N(t) = N₀ e^(kt)

• Population growth (k > 0)
• Radioactive decay (k < 0)
• Drug concentration in blood

Newton's Law of Cooling

dT/dt = -k(T - T_∞) → T(t) = T_∞ + (T₀ - T_∞)e^(-kt)

RC Circuit

R(dq/dt) + q/C = V₀ → Charge q(t) = CV₀(1 - e^(-t/RC))

Contaminant Decay

dc/dt + kc = 0 → c(t) = c₀ e^(-kt)

Summary

First-order ODEs model almost every rate process in engineering. Master separable equations and the integrating factor method. Always apply initial conditions to find the specific solution.