Solve Separable Differential Equation Calculator

Calculator

F(x) factor

G(y) factor

Initial x0

Initial y0

Target x

Step size

Decimal precision

Maximum rows

Accepted syntax

Example Data Table

Case	F(x)	G(y)	Initial condition	Expected form
Growth	x	y	y(0)=1	log(abs(y)) = x^2/2 + C
Power model	2*x	y^2	y(1)=2	-1/y = x^2 + C
Constant y factor	sin(x)	3	y(0)=4	y/3 = -cos(x) + C

Formula Used

A separable differential equation is written as:

dy/dx = F(x)G(y)

Separate variables:

dy/G(y) = F(x) dx

Integrate both sides:

∫ 1/G(y) dy = ∫ F(x) dx + C

With an initial condition y(x0)=y0:

C = H(y0) - R(x0)

Here, H(y) is the y-side integral, and R(x) is the x-side integral.

How to Use This Calculator

Enter the x factor in the F(x) field.
Enter the y factor in the G(y) field.
Add the initial condition values x0 and y0.
Enter the target x value and step size.
Select decimal precision and maximum table rows.
Press the calculate button to see the result above the form.
Use the CSV or PDF buttons to save the output.

Understanding Separable Equations

A separable differential equation has the derivative written as one x factor times one y factor. The usual form is dy/dx = F(x)G(y). This structure lets you move every y term to one side. It also moves every x term to the other side. After that, both sides can be integrated. The final answer is often implicit, because y may stay inside a logarithm, power, or trigonometric expression.

Why This Calculator Helps

Manual separation is easy to start, but errors appear quickly. A sign can be missed. A reciprocal can be copied wrongly. An initial value can change the constant. This calculator organizes those steps. It shows the separated statement, the integrated form, a constant rule, and a numerical table. The table is useful when the symbolic answer is hard to solve for y.

Advanced Options

The tool accepts common functions, powers, constants, and arithmetic. You can enter factors such as x^2+1, sin(x), exp(x), y, y^2, or 1+y. When an initial condition is supplied, the page builds a Runge Kutta table. It also compares a simple Euler estimate, so you can judge step size error. Smaller steps usually improve accuracy, but they create more rows.

Reading The Result

The implicit solution follows a clear pattern. The y side is the integral of one over G(y). The x side is the integral of F(x). The constant C joins the two sides. With x0 and y0, the constant is found by substitution. The numerical table then moves from x0 toward the target x value. Each row reports x, y, and slope data.

Good Input Practice

Use multiplication signs when needed. Write 3*x instead of 3x. Use log for natural logarithm. Use pi and e for constants. Avoid points where G(y) is zero, because division by zero may occur during separation. Check the step count before running a wide range. For classroom work, compare the displayed formula, table, CSV file, and PDF summary. Together, they make the solution easier to verify. Use the example table before entering new data. It shows acceptable syntax and expected behavior. This habit prevents format mistakes during longer homework sessions or tests online.

FAQs

What is a separable differential equation?

It is an equation where dy/dx can be written as F(x) times G(y). This allows x terms and y terms to be placed on separate sides before integration.

Can this calculator solve every differential equation?

No. It is designed for separable equations. If the equation cannot be written as F(x)G(y), another method may be needed.

Why is the answer sometimes implicit?

After integration, y may appear inside logs, powers, or other functions. In those cases, solving directly for y may not be simple.

What does F(x) mean?

F(x) is the part of the derivative that depends on x. Enter it using x, numbers, operators, and supported functions.

What does G(y) mean?

G(y) is the part of the derivative that depends on y. The calculator divides by this factor during the separation step.

Why do I need an initial condition?

An initial condition finds the constant C. It also lets the calculator create a numerical table from the starting point.

What step size should I use?

Use a smaller step for better numerical accuracy. Use a larger step for faster output and fewer rows.

Why can division by zero occur?

Separation uses dy divided by G(y). If G(y) equals zero near a table row, the real-valued calculation may stop.