Variable Separable Method: Solve Differential Equations

A first-order differential equation is variable-separable when it can be rearranged so that everything involving y sits with dy on one side and everything involving x sits with dx on the other. Then each side is integrated independently.

This is the first solution method every student meets because it turns a differential equation into two ordinary integrals. Population growth, radioactive decay, and many cooling and mixing problems all separate cleanly, making this method the workhorse of elementary modelling.

Key Concepts

• Separable form: dy/dx = f(x)·g(y), a product of a function of x alone and a function of y alone. Sums like f(x) + g(y) do not separate.
• Separation step: divide by g(y) and multiply by dx to get dy/g(y) = f(x) dx, valid wherever g(y) ≠ 0.
• One constant is enough: integrating both sides produces two constants, but they merge into a single C on one side.
• Implicit solutions: the result often cannot be solved explicitly for y; an equation relating x and y is an acceptable solution.
• Lost solutions: dividing by g(y) discards any constant solution with g(y) = 0, such as y = 0; these must be checked and added back if they satisfy the equation.

Formulas

• Separation: if dy/dx = f(x)·g(y) and g(y) ≠ 0, then ∫ dy/g(y) = ∫ f(x) dx + C.
• Exponential shortcut: dy/dx = ky separates to ln|y| = kx + C, giving y = Ae^(kx).

Worked Example

Solve dy/dx = 2xy with the initial condition y(0) = 3.

1. Separate: divide by y (y = 0 is not our solution since y(0) = 3) and multiply by dx: dy/y = 2x dx.
2. Integrate both sides: ∫ dy/y = ∫ 2x dx gives ln|y| = x² + C.
3. Exponentiate: |y| = e^(x² + C) = e^C·e^(x²). Writing A = ±e^C, we get y = Ae^(x²).
4. Apply y(0) = 3: 3 = A·e⁰ = A, so A = 3 and y = 3e^(x²).
5. Verify: dy/dx = 3·2xe^(x²) = 2x·(3e^(x²)) = 2xy, and y(0) = 3·1 = 3. Both check out.

Common Misconceptions

• Skipping the constant of integration. Without C the "solution" is just one curve, not the general solution, and no initial condition can be fitted.
• Adding C to both sides. Two constants collapse into one; writing C₁ and C₂ separately is redundant but harmless only if you merge them before finishing.
• Dividing by zero unnoticed. If g(y) = 0 has solutions, for example y = 0 in dy/dx = 2xy, check them: y = 0 indeed satisfies the equation and is a valid (equilibrium) solution.

Real-World Applications

• Population biology: dN/dt = kN separates to give exponential growth or decay of populations.
• Radioactivity: decay rates proportional to the remaining mass separate into the exponential decay law.
• Cooling: Newton's law of cooling separates to show temperature approaching the surroundings exponentially.
• Chemistry: first-order reaction rates integrate by separation to give concentration as a function of time.

Variable-separable equations surrender to a simple recipe: isolate y with dy, isolate x with dx, integrate both sides, and solve for the constant with the initial condition. Always check what dividing by g(y) may have discarded, and the method never fails you.