Practice Maths

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Separable Differential Equations

Key Terms

Separable DE
dy/dx = f(x)g(y) — the RHS factors into a function of x only times a function of y only.
Method
Rearrange to dy/g(y) = f(x) dx, then integrate both sides: ∫ dy/g(y) = ∫ f(x) dx + C.
General solution
contains the arbitrary constant C; a particular solution is found by applying an initial condition y(x⊂0;) = y⊂0;.
Exponential model
dy/dx = ky separates to give y = Aekx (A > 0 for growth, A < 0 for decay when k < 0).
After integrating
, always include +C on one side before solving for y; the constant absorbs any constants from both integrals.
Key Formulas:
  • Separation: ∫ dy/g(y) = ∫ f(x) dx + C
  • Exponential: dy/dx = ky ⇒ y = Cekx
  • Newton’s cooling: dT/dt = −k(T − Ta) ⇒ T − Ta = Ae−kt
  • Logistic (separable via partial fractions): dP/dt = rP(1 − P/K)
Hot Tip Always include +C immediately after integrating — before rearranging for y. The constant C on one side absorbs any constants from both integrals. Never substitute numerical values before differentiating in related-rate style problems.

Worked Example 1 — Separable with Initial Condition

Solve dy/dx = 2xy, given y(0) = 3.

Step 1 — Separate: dy/y = 2x dx

Step 2 — Integrate: ln|y| = x² + C

Step 3 — Solve for y: y = Aex² (where A = ±eC)

Step 4 — Apply IC: y(0) = A = 3, so y = 3ex²

Worked Example 2 — Power-type Separable

Solve dy/dx = y².

Separate: dy/y² = dx ⇒ y−2 dy = dx

Integrate: −1/y = x + C

Solve: y = −1/(x + C)

What Makes a Differential Equation Separable?

A first-order differential equation dy/dx = F(x, y) is called separable if F(x, y) can be written as a product f(x) × g(y) — a function of x only multiplied by a function of y only. This allows us to “separate” the variables onto opposite sides of the equation.

Examples of separable DEs: dy/dx = 3xy, dy/dx = exy², dy/dx = (x + 1)/y. Non-separable: dy/dx = x + y.

The Method of Separation of Variables

Step 1 — Identify and separate: Write the DE as g(y)−1 dy = f(x) dx, moving all y-terms (including dy) to the left and all x-terms (including dx) to the right.

Step 2 — Integrate both sides: ∫ dy/g(y) = ∫ f(x) dx. Add only one constant of integration C on the right-hand side.

Step 3 — Solve for y (if possible) to write the general solution explicitly.

Step 4 — Apply initial condition (if given) to find the value of C and hence the particular solution.

Exponential Growth and Decay

The DE dy/dx = ky is the most important separable equation. Separating: dy/y = k dx. Integrating: ln|y| = kx + C. Exponentiating: y = Aekx where A = ±eC absorbs the sign.

If k > 0, we have exponential growth; if k < 0, exponential decay. The constant A equals the initial value y(0).

Newton’s Law of Cooling

An object cools (or warms) toward the ambient temperature Ta at a rate proportional to the temperature difference: dT/dt = −k(T − Ta), where k > 0. Let u = T − Ta; then du/dt = −ku, giving u = u⊂0;e−kt, so T = Ta + (T0 − Ta)e−kt.

Mixing Problems

For a tank with volume V, inflow rate rin, concentration cin, and outflow matching inflow: if x(t) = amount of substance at time t, then dx/dt = rate in − rate out = rin×cin − (x/V)×rout. This is often a first-order linear DE or, in the simple case of pure water flushing, separable.

Logistic Growth

dP/dt = rP(1 − P/K) is separable. Separating and using partial fractions: ∫ dP/[P(K−P)] = (r/K)t + C. This gives the S-shaped logistic curve P(t) = K/(1 + Ae−rt).

Exam Tip: Always write the constant as +C immediately after integrating — never skip it or “add it later”. The most common error is forgetting C until after substituting the initial condition.
Exam Tip: When ln|y| appears, the general solution is y = Aef(x) where A can be positive or negative (absorbing ±). In many physical problems y > 0, so A > 0.

Mastery Practice

  1. Solve dy/dx = 3y. Fluency

  2. Solve dy/dx = x/y with initial condition y(0) = 2. Fluency

  3. Solve dy/dx = xy². Fluency

  4. A population satisfies dP/dt = 0.05P with P(0) = 1000. Find P(t). Fluency

  5. Solve dy/dx = (x + 1)/(y − 1). Understanding

  6. Solve dy/dx = ex − y. Understanding

  7. Newton’s cooling: dT/dt = −k(T − 20). A cup of coffee starts at 90°C and cools to 60°C after 5 minutes. Find T(t) and estimate the temperature after 10 minutes. Understanding

  8. Solve dy/dx = y sin(x) with y(0) = 2. Understanding

  9. Logistic growth: dP/dt = 0.002P(1000 − P) with P(0) = 100. Use partial fractions to find P(t). Problem Solving

  10. A 50 L tank contains 5 kg of dissolved salt. Pure water flows in at 2 L/min and the well-mixed solution drains at 2 L/min. Let x be the kg of salt at time t (minutes). Set up and solve the DE, then find when x = 1 kg. Problem Solving

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