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First Order Differential Equations

Learn to recognize and solve first-order ODEs (separable, linear, Bernoulli, homogeneous) with worked examples and physics applications.

  • University Physics Year 1
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First-order ordinary differential equations (ODEs) are equations for an unknown function y(x) that involve only its first derivative dy/dx.

They show up constantly in Year 1 physics, e.g. motion with drag, RC circuits, and exponential relaxation processes.

Why physics needs this (examples you can click)
What you’re usually trying to do
  • Identify the ODE type (separable / linear / Bernoulli / …)
  • Solve for y(x) (or v(t), q(t), etc.)
  • Use initial conditions to determine constants (e.g. y(0) = y₀)
  • Check units and limiting cases (small t, large t, etc.)

Start here (method selection)

If you’re not sure what to try first, work through this quick checklist (in order):

  1. Write it in a clean “derivative equals something” form

    dy/dx = f(x,y)

  2. Test for separable (can you split into a pure-x side and a pure-y side?)

    dy/dx = f(x) g(y) ⇒ 1/g(y) dy = f(x) dx

  3. Test for linear

    dy/dx + P(x) y = Q(x)

  4. Test for Bernoulli

    dy/dx + P(x) y = Q(x) yⁿ (n ≠ 0,1)

  5. Test for homogeneous (first-order) (depends only on y/x)

    dy/dx = F (y/x)

  6. (Optional) Test for exactness

    M(x,y) dx + N(x,y) dy = 0

Printable checklist: Steps for solving first-order differential equations

Core methods (with worked pages)

  • Separable ODEs

    Rearrange into g(y) dy = f(x) dx, then integrate both sides.

    Open lesson

  • First-order linear ODEs

    Standard form y' + P(x)y = Q(x) and the integrating factor method.

    Open lesson

  • Bernoulli ODEs

    Convert to linear with the substitution u = y¹⁻ⁿ.

    Open lesson

  • Homogeneous first-order ODEs

    Use y = vx (or x = vy) to reduce it to separable form.

    Open lesson

Method templates (copy/paste checklists)

Separable

If you can rewrite the ODE as

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

then separate variables and integrate:

1/g(y) dy = f(x) dx.

Linear (integrating factor)

Write

y' + P(x) y = Q(x).

Then

μ(x) = e^(∫ P(x) dx), (μ y)' = μ Q, y = (1/μ)(∫ μ Q dx + C).

Bernoulli

If

y' + P(x)y = Q(x)yⁿ (n ≠ 0,1),

divide by yⁿ and substitute u = y¹⁻ⁿ to reduce to a linear ODE in u.

Relaxation-to-equilibrium (physics form)

If

dy/dt = -(1/τ)(y-y_∞),

then the solution is always

y(t) = y_∞ + (y₀-y_∞)e^(-t/τ).
Pitfalls to watch for
  • If you divide by a factor like g(y), you may lose equilibrium solutions where that factor is zero.
  • If your solution contains logs, remember absolute values: ln |y|.
  • Always apply the initial condition(s). Most physics problems expect C eliminated.

A worked example you’ll see everywhere: exponential relaxation

Many “approach-to-equilibrium” models have the form:

dy/dt = -(1/τ)(y-y_∞),

where τ is a time constant and y_∞ is the long-time (steady-state) value.

This is a first-order linear ODE. Separate variables (or use integrating factors):

dy/(y-y_∞) = -dt/τ.

Integrate and apply y(0) = y₀:

y(t) = y_∞ + (y₀-y_∞)e^(-t/τ).

Interpretation:

  • At t = τ, the “gap to equilibrium” drops to e⁻¹ ≈ 0.37 of its initial size.
  • As t → ∞, y(t) → y_∞.

Worked example (physics): RC discharge in one line

Discharging a capacitor through a resistor gives:

dq/dt = -(1/RC)q, q(0) = q₀.

This is separable:

1/q dq = -1/RC dt ⇒ ln |q| = -t/RC + C₁ ⇒ q(t) = q₀e^(-t/RC).

The time constant is τ = RC. For the full setup/sign convention, see UY1: RC Circuits.

Worked example (physics): linear drag and terminal speed

For a falling object with linear drag (downward positive),

mdv/dt = mg-kv, v(0) = 0.

Rewrite as a linear ODE:

dv/dt + (k/m)v = g.

The solution is:

v(t) = vₜ(1-e^(-t/τ)), vₜ = mg/k, τ = m/k.

This makes the physics obvious: v(t) approaches the terminal speed vₜ exponentially on the timescale τ.

For a full discussion (including quadratic drag), see UY1: Resistive Forces.

Physics applications (see the ODE in action)

Practice (with hints + answers)

1) Classify (no solving needed): y' + 4y = x^2

Hint: Does it match y' + P(x)y = Q(x)?

Answer: Linear.

2) Classify: y' = x y^3

Hint: Can you separate into f(x)g(y)?

Answer: Separable (also Bernoulli if rewritten, but separation is fastest):

dy/y³ = x dx.
3) Solve: y' = -2y, with y(0)=5

Hint: Exponential decay.

Answer:

y(t) = 5e^(-2t).
4) Solve: y' + y = 3, with y(0)=0

Hint: Use integrating factor (or recognize relaxation to y_∞ = 3 with τ = 1).

Answer:

y(t) = 3(1-e^(-t)).
5) Find equilibrium (constant) solutions: dy/dt = (y-1)(y+2)

Hint: Set the right-hand side to zero.

Answer: y(t) = 1 and y(t) = -2 are equilibrium solutions.

6) Quick checks: what limits should you test for y(t)=y_infty+(y0-y_infty)e^{-t/tau}?

Hint: Think t → 0 and t → ∞.

Answer:

  • t → 0: y(0) = y₀.
  • t → ∞: y → y_∞.

Quick reference (forms you should recognize)

Separable

dy/dx = f(x) g(y)

Linear

dy/dx + P(x) y = Q(x)

Bernoulli

dy/dx + P(x) y = Q(x) yⁿ

Homogeneous (first-order)

dy/dx = F (y/x)

Common mistakes to avoid

  • Algebra slip while “separating” (you must end with a pure-x side and a pure-y side).
  • Forgetting the constant of integration (or losing it when simplifying).
  • Forgetting absolute values in logs (e.g. ∫ dx/x = ln |x| + C).
  • Not applying initial conditions (most physics problems expect a fully determined y(x)).

Next (second-order ODEs)

Many oscillation and motion problems lead to second-order ODEs:

Back to Mathematics for Undergraduate Physics