Separation of Variables

Unit 7 · Differential Equations

Separation of Variables

A separable DE has the form dy/dx = g(x)·h(y). Separate by writing dy/h(y) = g(x) dx, integrate both sides, then solve for y. The arbitrary constant C appears after integration; use an initial condition to find the particular solution.

Separation of Variables Steps

1. Separate

\frac{dy}{h(y)} = g(x)\,dx

2. Integrate both sides

\int\frac{dy}{h(y)} = \int g(x)\,dx + C

3. Solve for y; apply initial condition if given

Exponential Growth/Decay

DE and solution

\frac{dy}{dt} = ky \implies y = Ce^{kt}

Half-life (k > 0 decay)

t_{1/2} = \frac{\ln 2}{k}

Caution: Write ln|y|, not ln y — the absolute value matters when solving for y. Also, apply initial conditions after finding the general solution, not before.

Type 1

Finding the General Solution

Separate variables, integrate both sides, and express y in terms of x with an arbitrary constant.

Example 1

Find the general solution of dy/dx = x²/y.

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Type 2

Particular Solution with Initial Condition

After finding the general solution, substitute the initial condition to determine C explicitly.

Example 2

Solve dy/dx = −2y, y(0) = 5.

Example 3

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

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Type 3

Exponential Growth/Decay and Newton's Law of Cooling

Recognize dP/dt = kP or dT/dt = −k(T−T∞) as separable DEs, solve, and apply initial/contextual conditions.

Example 4

Bacteria double every 3 hours. Initial count: 500. Write the DE, solve, and find P(9).

Example 5

A substance decays with dA/dt = −0.03A, A(0) = 200 g. Find A(t) and when 100 g remains.

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