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Steps for Solving First-Order Differential Equations

A quick checklist to identify the type of a first-order ODE (linear, Bernoulli, separable, homogeneous, exact) before solving.

  • University Physics Year 1
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This is a quick method-selection checklist. In physics, you often derive a first-order ODE from:

  • Newton’s 2nd law written as a first-order equation for v(t)
  • Kirchhoff’s laws written as an equation for q(t) or i(t)
  • “Relaxation to equilibrium” models with a time constant

and the main speed boost is recognizing the type fast so you don’t try the wrong tool.

Why physics needs this (examples you can click)

Steps

  1. Rewrite into a standard form

    dy/dx = f(x,y)

  2. Test for separable (usually the fastest win)

    Can you rearrange to a pure-y side times dy and a pure-x side times dx?

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

    If yes: integrate both sides, then apply initial conditions.

  3. Test for linear (the most common physics pattern)

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

    If yes: use an integrating factor. See First-order linear ODEs.

  4. Test for Bernoulli (a “linear after substitution” type)

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

    If yes: substitute u = y¹⁻ⁿ to reduce to linear. See Bernoulli ODEs.

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

    dy/dx = F (y/x)

    If yes: use the substitution y = vx (or x = vy) to make it separable.

  6. (Optional) Test for exactness (often introduced later)

    Exact equations are often written as:

    M(x,y) dx + N(x,y) dy = 0
Pitfall: don’t force an equation into the wrong type
  • If separating variables leaves mixed x and y on both sides, it is not separable (at least not directly).
  • “Looks linear” is not enough: you must be able to rewrite it as y' + P(x)y = Q(x) (no powers like y² on the left).
  • If you do a substitution (e.g. y = vx), always rewrite y' carefully: y' = v + xv'.
After you solve

Always apply the initial condition(s) to fix constants, then check:

  • Units/dimensions
  • Limiting behavior (e.g. t → 0 and t → ∞)
  • Whether the answer stays physical in the domain you care about

Quick classification examples (what you’d do first)

  1. RC charging form

    dq/dt + (1/RC)q = E/R

    This is linear in q(t). First move: identify P(t) = 1/RC, compute μ(t) = e^(∫ P dt) = e^(t/RC).

  2. Quadratic drag (downward motion, v ≥ 0)

    dv/dt = g-(b/m)v²

    This is separable. First move: dv/(g-(b/m)v²) = dt.

  3. Bernoulli type

    y' + y = x y²

    This is Bernoulli (n = 2). First move: divide by y² then substitute u = 1/y.

  4. Homogeneous first-order

    dy/dx = (y/x)² + y/x

    This depends only on y/x. First move: set y = vx, so y' = v + xv'.

Practice: choose the method (with hints + answers)

1) Classify and pick a first move: dT/dt = -k (T - T_env)

Hint: Can you rewrite it as T' + P T = Q?

Answer: Linear (also separable). Standard relaxation form:

dT/dt + k_PT = kTₑₙᵥ_Q.
2) Classify: y' + (2/x) y = x^2

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

Answer: Linear. Integrating factor:

μ(x) = e^(∫ 2/x dx) = e^(2 ln |x|) = x².
3) Classify (physics): dv/dt = g - (b/m) v|v|

Hint: In many UY1 “falling” problems you can assume v ≥ 0 after release.

Answer: Separable. If v ≥ 0 then v|v| = v² and

dv/(g-(b/m)v²) = dt.

If the motion can reverse, keep v|v| and track the sign carefully.

4) Classify: y' + y = x y^3

Hint: Compare with y' + P(x)y = Q(x)yⁿ.

Answer: Bernoulli (n = 3). Divide by y³ and use u = y¹⁻³ = y⁻².

5) Classify: y' = sin(y/x)

Hint: The right-hand side depends only on y/x.

Answer: Homogeneous first-order. Use y = vx, so y' = v + xv', giving

v + xv' = sin(v) ⇒ dv/(sin v - v) = dx/x.

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