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The Separable DE Calculator solves differential equations of the form $$\frac{dy}{dx} = f(x) \cdot g(y)$$ where the right-hand side factors into a function of $$x$$ alone multiplied by a function of $$y$$ alone. Separable equations are among the most common and solvable types of first-order ODEs, and this calculator handles five standard forms with known closed-form solutions.
The method of separation of variables rewrites the equation as $$\frac{dy}{g(y)} = f(x)\,dx$$ and integrates both sides independently. The key requirement is that the right-hand side must factor into a product of a function of $$x$$ and a function of $$y$$ — hence the name "separable."
The five equation types available are: (1) Exponential $$dy/dx = ky$$, the classic growth/decay model; (2) Power/Reciprocal $$dy/dx = ky^2$$, which can exhibit finite-time blow-up; (3) Square Root $$dy/dx = k\sqrt{y}$$, modeling processes where the rate depends on the square root of the quantity; (4) Logistic $$dy/dx = ky(M-y)$$, the density-dependent growth model; (5) Product $$dy/dx = kxy$$, where the rate depends on both position and the current value.
The Power equation $$dy/dx = ky^2$$ is particularly interesting because its solution $$y = 1/(1/y_0 - kx)$$ develops a singularity at $$x^* = 1/(ky_0)$$ where $$y \to \infty$$ in finite time. This phenomenon, called blow-up, occurs in combustion theory, nonlinear optics, and certain chemical reactions.
The Square Root equation $$dy/dx = k\sqrt{y}$$ separates to $$y^{-1/2}dy = k\,dx$$, giving $$2\sqrt{y} = kx + 2\sqrt{y_0}$$, hence $$y = (\sqrt{y_0} + kx/2)^2$$. This models diffusion-limited reactions and certain absorption processes.
The Product equation $$dy/dx = kxy$$ separates to $$dy/y = kx\,dx$$, giving $$\ln y = kx^2/2 + C$$, hence $$y = y_0 e^{kx^2/2}$$. This Gaussian-type growth appears in probability theory and quantum mechanics.
Select an equation type, enter the rate constant and initial condition, then evaluate at your chosen point. The calculator provides the solution value, instantaneous rate, solution form, and (for the power equation) the blow-up point.
Each equation is solved by separating variables and integrating.
Exponential ($$dy/dx = ky$$): $$\int dy/y = \int k\,dx \implies \ln|y| = kx + C \implies y = y_0 e^{kx}$$
Power ($$dy/dx = ky^2$$): $$\int y^{-2}dy = \int k\,dx \implies -1/y = kx + C \implies y = \frac{1}{1/y_0 - kx}$$
Singularity at $$x^* = \frac{1}{ky_0}$$ where the denominator reaches zero.
Square Root ($$dy/dx = k\sqrt{y}$$): $$\int y^{-1/2}dy = \int k\,dx \implies 2\sqrt{y} = kx + 2\sqrt{y_0} \implies y = \left(\sqrt{y_0} + \frac{kx}{2}\right)^2$$
Logistic ($$dy/dx = ky(M-y)$$): Using partial fractions: $$y(x) = \frac{M}{1 + Ae^{-kMx}}$$ where $$A = \frac{M - y_0}{y_0}$$
Product ($$dy/dx = kxy$$): $$\int dy/y = \int kx\,dx \implies \ln|y| = kx^2/2 + C \implies y = y_0 e^{kx^2/2}$$
y(x) is the solution value at the specified point. The behavior varies dramatically by equation type — exponential growth, blow-up, quadratic growth, sigmoidal saturation, or Gaussian growth.
dy/dx at x is the instantaneous rate of change, computed by substituting y(x) back into the original equation.
Solution Form shows the analytical formula being used, so you can verify and use it for further analysis.
Blow-up / Singularity x* is shown for the power equation (dy/dx = ky²). At this point, y → ∞ in finite time. If x is close to or past x*, the calculator may return very large values. For other equation types, this is zero (no finite-time singularity).
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With y₀ = 10 and k = 0.5, the blow-up occurs at x* = 1/(0.5·10) = 0.2. At x = 0.15: y = 1/(0.1 - 0.075) = 1/0.025 = 40. The solution has quadrupled from 10 to 40 with only 75% of the way to blow-up. The rate dy/dx = 0.5·40² = 800 is extremely high.
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y(3) = 5·e^(0.5·9/2) = 5·e^(2.25) ≈ 5·9.4877 ≈ 47.44. The Gaussian growth accelerates dramatically as x increases because the exponent grows quadratically.
A first-order ODE dy/dx = F(x,y) is separable if F(x,y) can be written as f(x)·g(y) — a product of a function of x alone and a function of y alone. This allows us to 'separate' the variables: dy/g(y) = f(x)dx, and integrate each side independently.
Finite-time blow-up occurs when the solution goes to infinity at some finite x value. The equation dy/dx = ky² with k > 0 and y₀ > 0 blows up at x* = 1/(ky₀). Near this point, y grows so rapidly that it becomes infinite. This models thermal runaway, certain chemical explosions, and gravitational collapse.
No. Many first-order ODEs cannot be separated into f(x)·g(y) form. For example, dy/dx = x + y is not separable. Non-separable equations may require other methods: integrating factors, exact equations, substitution, or numerical methods.
With dy/dx = ky, the growth rate is proportional to y — as y grows, it grows faster, but only linearly faster. With dy/dx = ky², the growth rate is proportional to y² — a positive feedback that accelerates super-exponentially, overwhelming the solution in finite time.
For the exponential and product types, y₀ = 0 gives the trivial solution y(x) = 0 for all x. For the power and logistic types, y₀ = 0 is a singularity in the solution formula (division by zero). For the square root type, y₀ = 0 gives y = (kx/2)², which is valid.
They are the same equation. Writing dy/dx = ky(M-y) emphasizes the separable form where g(y) = y(M-y). Expanding gives dy/dx = kMy - ky², equivalent to the standard logistic dy/dx = ry(1 - y/K) with r = kM and K = M.
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