Homogeneous Differential Equation Solver Calculator

Calculator Inputs

This tool solves first-order homogeneous equations in the form below.

dy/dx = (a·x + b·y) / (c·x + d·y)

Coefficient a

Coefficient b

Coefficient c

Coefficient d

Initial x₀

Initial y₀

Target x

Step size

Example Data Table

Example equation: dy/dx = (x + y) / x, with y(1) = 0. Exact solution: y = x ln(x).

x	Exact y = x ln(x)	dy/dx = 1 + ln(x)	v = y/x = ln(x)
1.0000	0.0000	1.0000	0.0000
1.5000	0.6082	1.4055	0.4055
2.0000	1.3863	1.6931	0.6931

Formula Used

For the homogeneous model dy/dx = (a·x + b·y) / (c·x + d·y), use the substitution y = v·x.

Then dy/dx = v + x·dv/dx.

Substituting gives:

v + x·dv/dx = (a + b·v) / (c + d·v)

Rearranging gives the reduced separable equation:

x·dv/dx = [a + (b − c)v − d·v²] / [c + d·v]

So the separated form becomes:

(c + d·v) / [a + (b − c)v − d·v²] · dv = dx / x

This page evaluates the curve numerically with fourth-order Runge–Kutta after transforming the equation and applying the chosen initial condition.

How to Use This Calculator

Enter coefficients a, b, c, and d for the equation model.
Provide the initial point (x₀, y₀).
Choose the target x value where you want the estimate.
Set a practical step size. Smaller values improve detail.
Click Solve Equation to generate the numerical solution.
Review the transformed equation, summary cards, graph, and full table.
Use CSV or PDF export to save results for notes, homework, or revision.

FAQs

1) What is a homogeneous differential equation?

A first-order differential equation is homogeneous when its slope depends on a ratio such as y/x, letting substitution reduce it to a separable form.

2) Which equations does this calculator handle?

This calculator handles equations written as dy/dx = (a·x + b·y) / (c·x + d·y). That family is homogeneous because numerator and denominator have the same degree.

3) Why must the interval avoid x = 0?

The substitution y = v·x uses v = y/x, so x cannot be zero. Crossing zero also conflicts with the separated form containing dx/x.

4) Does the tool return exact or numerical solutions?

It shows the transformation steps and then computes a numerical solution with Runge–Kutta. That approach is reliable for graphing, tables, and target-point estimates.

5) What does the substitution y = v·x do?

It converts the original dependent variable into a ratio variable. That simplification exposes separability and makes the equation much easier to analyze.

6) Why might the solver stop during calculation?

The slope formula fails when c·x + d·y becomes zero. That creates a singular point, so the numerical method cannot proceed safely through it.

7) What do CSV and PDF exports include?

The exports capture the computed solution table. The PDF also includes the result summary and graph for clearer documentation.

8) Can I solve on negative x values?

Yes, provided the whole interval stays negative or positive. The path should not cross zero, and denominator singularities must still be avoided.