What Is a Separable Differential Equation?
A first-order equation is separable when its right-hand side factors into a pure-x part times a pure-y part: y′ = f(x)·g(y). That structure permits the oldest solving trick in the subject, put every y on one side, every x on the other, and integrate both:
dy / g(y) = f(x) dx ⟹ ∫ dy/g(y) = ∫ f(x) dx + C
Two ordinary integrals replace the differential equation entirely. The answer arrives as an implicit relation G(y) = F(x) + C, sometimes solvable for y explicitly, sometimes best left as is, with the constant C pinned down by an initial condition. Separation of variables is where nearly every differential equations course begins, because it converts the new problem (an ODE) into the previous chapter's problem (antiderivatives), and because an outsized share of nature's first-order laws happen to be separable.
How to Use This Calculator
Enter the two factors f(x) and g(y). The calculator separates the equation, integrates each side symbolically with its rule-based engine, and presents the implicit general solution with C explicit. Provide an initial condition and C is evaluated for you, yielding the particular solution, plotted as the curve through your point via high-accuracy numerical integration (so the graph is honest even when the implicit relation resists solving for y). If a factor is beyond the symbolic integrator, the error message says so and points you to the numeric tools.
Worked Example
Solve y′ = xy with y(0) = 1. Separate: dy/y = x dx. Integrate both sides: ln|y| = x²/2 + C. Exponentiating gives y = Ae^(x²/2), and y(0) = 1 forces A = 1:
y = e^(x²/2)
Note what the initial condition did: it selected one curve from an infinite family parameterized by A. The fourth chip runs the mirror problem y′ = −2xy, whose solution e^(−x²) is the Gaussian bell, the same separation, one sign flipped, and growth becomes the most famous decay curve in statistics.
The Singular-Solution Subtlety
Dividing by g(y) is legal only where g(y) ≠ 0, and each root g(c) = 0 is itself a constant solution y = c that the division silently discards. For y′ = xy, the constant y = 0 is a perfectly good solution not produced by any finite C (it's the A = 0 case that exponentiation seemed to forbid, since A = e^C is "never zero", until one remembers the lost solution). Textbook grading and real modeling both punish this omission: the logistic equation's equilibria, a cooling body already at ambient temperature, a population starting at zero, all are equilibrium solutions found by solving g(y) = 0 before separating. Always list them alongside the separated family.
Common Mistakes to Avoid
- Separating the inseparable. y′ = x + y cannot be written f(x)·g(y), no algebra will split a sum into a product. (It's linear instead; different technique.) Check the factored form honestly before dividing.
- Adding C to both sides. One constant suffices: the two integration constants merge into a single C on one side. Carrying two just breeds algebra errors when applying the initial condition.
- Dropping the absolute value in ln|y| too early. The safe route: keep |y| until exponentiation, let A absorb the sign, then use the initial condition to fix A's sign. Solutions can't cross y = 0 anyway (it's an equilibrium), so the sign of y is locked by y₀.
- Assuming solutions live forever. The second chip's y′ = y², y(0) = 1 solves to y = 1/(1 − x), which explodes at x = 1, finite-time blow-up from an innocent-looking equation. The interval of validity is part of the answer, not a footnote.
Real-World Applications
Separable equations are the modeling canon: exponential growth and decay (y′ = ky) covers radioactive dating, compound interest, and first-order drug elimination; Newton's law of cooling (T′ = −k(T − Tamb)) times everything from forensic time-of-death estimates to coffee; the logistic equation caps growth at carrying capacity for populations, epidemics, and market adoption; and Torricelli's law drains tanks in every fluid-mechanics course. Each is separable, and each is solved in practice exactly as this page does it.
Beyond the classics, chemists integrate separable rate laws to extract reaction orders from concentration data; atmospheric scientists derive the barometric pressure formula by separating dP/dh = −ρg with the ideal gas law; and electrical engineers solve RC charge/discharge separably before graduating to Laplace transforms. Separation is also the conceptual gateway to the PDE technique of the same name, solving the heat and wave equations by splitting variables, so the reflex trained here pays off far beyond first-order ODEs.
Frequently Asked Questions
How do I tell if an equation is separable?
Try to write the right side as (function of x alone) × (function of y alone). Products, quotients, and things like e^(x+y) = eˣ·e^y separate; genuine sums like x + y do not. When in doubt, test whether f(x,y)·f(1,1) = f(x,1)·f(1,y) for the candidate factorization, factorable functions pass.
Why is the answer sometimes left implicit?
Because solving G(y) = F(x) + C for y may be impossible in elementary terms (try y + ln y = x² + C). The implicit relation still defines the solution curve completely, and for graphing, evaluating, or checking initial conditions, implicit is fully usable, as this page's plot demonstrates.
Is "multiplying both sides by dx" legitimate?
It's shorthand for a rigorous move: writing (1/g(y))·(dy/dx) = f(x) and integrating both sides in x, with the left side transformed by the chain rule (integration by substitution). The differentials notation compresses that argument and always gives the same answer, Leibniz designed it to.
What if g(y₀) = 0 at my initial condition?
Then your solution is the equilibrium y = y₀, constant forever, no separation needed. (With smooth g that equilibrium is unique through the point; with non-smooth g like y^(1/3), uniqueness can genuinely fail, which is the classic counterexample territory of the existence-uniqueness theorem.)
How does the initial condition determine the interval of validity?
The solution extends from x₀ until it hits a barrier: a blow-up (like x = 1 for y′ = y²), a point where the formula leaves the domain (log of a negative), or an equilibrium it approaches but never reaches. The valid interval is always the one containing x₀ solutions don't jump barriers.
Which is better here: this symbolic solver or Euler/RK4?
Symbolic when it works, exact formulas reveal structure (half-lives, blow-up times, asymptotes) that numbers only hint at. Numeric when the integrals resist or the equation isn't separable at all. Professionals use both: formula for insight, integrator for everything else.