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Calculus · Differential equations

Separable Differential Equation Calculator

Solve selected first-order separable initial-value problems and evaluate the analytical solution at a target x.

Formula shown Runs locally Reviewed Aug 11, 2026
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Equation guide

Controlled first-order initial-value problems with separable variables

Separate

dy/g(y)=f(x)dx

Integrate

∫dy/g(y)=∫f(x)dx+C

Initial condition

y(x₀)=y₀ determines C
Calculus · Differential equations

Solve a controlled separable initial-value problem

The initial condition is y(x₀)=y₀. Solutions are real and evaluated only where their analytical expression remains finite.

Inputs and displayed decimal results use at most three decimal places. Full internal precision is retained during calculation.

dy/g(y)=f(x)dx

Choose a separable model and provide one initial condition.

Using the model

Symbols, assumptions and limitations

Interpretation notes

The supported models are y′=ky, y′=kxy, and y′=ky². Each uses a real initial condition, and evaluation is rejected at singularities or when the result leaves the finite numeric range.

Worked examples

See the method in practice

01

Exponential growth

For y′=0.5y with y(0)=2, the solution is y=2e^(0.5x).

02

Finite-time singularity

For y′=y² with y(0)=1, the solution 1/(1−x) diverges at x=1.

Questions

Frequently asked

What makes an equation separable?

Its differential terms can be rearranged so all y-dependent factors accompany dy and all x-dependent factors accompany dx.

Why is an initial condition needed?

It determines the integration constant and selects one solution from the family.

Why can a solution fail at a finite x?

Some nonlinear models have denominators that reach zero, producing a genuine singularity.

Sources & review

Equations and examples are checked against the references below. Results are educational and should be independently verified for safety-critical work.

OpenStax Calculus Volume 2 — Separable Equations

Written by the STEM Hub editorial team · Reviewed August 11, 2026 · Review methodology