Separable Differential Equation Calculator

Enter your separable differential equation (e.g. dy/dx = x*y) and optional initial condition (e.g. y(0)=1) to get the general or particular solution. The Separable Differential Equation Calculator separates variables, integrates both sides, and returns the solved expression for y in terms of x — with each step shown clearly.

Select a standard separable ODE form to solve.

Used only when dy/dx = k·y is selected. Typical values: 0.05 for slow growth, −0.1 for decay.

The x value for the initial condition, e.g. 0 for y(0)=C.

The value of y at x₀, e.g. y(0)=1.

Point at which to numerically evaluate the solution y(x).

Results

y(x) at Evaluated Point

--

Solution Type

--

Integration Constant C

--

Step 1 — Separate Variables

--

Step 2 — Integrate Both Sides

--

Step 3 — General Solution

--

Step 4 — Particular Solution

--

Solution Curve y(x)

Results Table

Frequently Asked Questions

What is a separable differential equation?

A separable differential equation is a first-order ODE that can be written in the form dy/dx = f(x)·g(y), where the right-hand side factors into a function of x alone multiplied by a function of y alone. This allows you to separate variables — moving all y terms to one side and all x terms to the other — before integrating each side independently.

How do you solve a separable ODE step by step?

First, rewrite the equation so all y terms (including dy) are on the left and all x terms (including dx) are on the right. Second, integrate both sides. Third, solve for y explicitly if possible to get the general solution. Finally, if an initial condition y(x₀) = y₀ is given, substitute it to find the constant of integration C and obtain the particular solution.

What is the constant of integration C?

When you integrate both sides of a separated ODE, an arbitrary constant C appears. The general solution contains C and represents a family of infinitely many solution curves. When an initial condition is provided, you substitute the known point (x₀, y₀) into the general solution and solve for the specific numerical value of C, yielding a unique particular solution.

What is the difference between a general solution and a particular solution?

The general solution contains an arbitrary constant C and describes the entire family of curves satisfying the ODE. A particular solution is obtained by applying an initial condition — a known value of y at a specific x — which pins down C to a single number, giving one specific curve from the family.

Can every first-order ODE be solved by separation of variables?

No. Separation of variables only works when the ODE can be factored as dy/dx = f(x)·g(y). Many first-order ODEs are not separable and require other methods such as integrating factors (for linear ODEs), exact equation techniques, or Bernoulli substitution.

What does dy/dx = k·y model in real life?

This classic separable ODE models exponential growth and decay. When k > 0, it describes population growth, compound interest, or bacterial reproduction. When k < 0, it models radioactive decay, cooling (Newton's Law of Cooling), or drug elimination from the bloodstream. The solution is always y = C·e^(kx).

Why does the calculator show numerical values instead of symbolic expressions?

This calculator evaluates the solution numerically at a user-specified point for clarity and quick verification. Symbolic algebra engines (like Wolfram Alpha or Symbolab) can display full LaTeX expressions, but this tool focuses on showing you the solution steps conceptually and then computing exact numeric outputs so you can immediately check specific values.

What happens if the initial condition leads to division by zero or a log of a negative number?

Some initial conditions fall outside the domain of the solution. For example, dy/dx = y² leads to y = 1/(C−x), which has a vertical asymptote. If your chosen x₀ or evaluation point is at or beyond that asymptote, the solution is undefined. The calculator will flag such cases and return a warning rather than an incorrect value.

More Math Tools