Homogeneous Differential Equation Calculator

Compute homogeneous equation solutions with steps and graphs. Export clean reports and understand substitutions through clear examples for smarter practice.

Calculator

This advanced calculator solves the common homogeneous family dy/dx = k(y/x)n. It applies the substitution y = vx, evaluates the solution at a chosen x value, builds a plot table, and prepares export-ready output.

Plotly Graph

The graph displays the solution curve y(x) generated from the chosen constants and plotting interval.

Example Data Table

# x y(x) v = y/x dy/dx
1 0.5 0.612372 1.224744 1.5
2 1 1.414214 1.414214 2
3 1.5 2.12132 1.414214 2
4 2 2.828427 1.414214 2
5 2.5 3.535534 1.414214 2

This table is exportable and updates after each calculation.

Formula Used

Homogeneous form:
dy/dx = F(y/x)
Substitution:
y = vx, so dy/dx = v + x(dv/dx)
Supported family:
dy/dx = k(y/x)n
For n ≠ 1:
y = x[k + Cxn-1]1/(1-n)
For n = 1:
y = Cxk

How to Use This Calculator

  1. Enter the coefficient k and exponent n for the homogeneous family.
  2. Choose the integration constant C that defines one solution curve.
  3. Provide the x value where you want the calculator to evaluate y(x) and dy/dx.
  4. Set the graph interval using plot start and plot end.
  5. Pick how many points the curve should include.
  6. Click Solve Equation to display the result above the form.
  7. Use the export buttons to save the generated table as CSV or PDF.
  8. Review the steps and graph to understand the substitution process clearly.

FAQs

1. What is a homogeneous differential equation?

It is a first-order differential equation where the right side can be written only in terms of y/x. That structure allows the substitution y = vx, which simplifies the equation into a separable or linear form.

2. Why does the substitution y = vx work?

Because homogeneous equations depend on the ratio y/x. Replacing y with vx turns that ratio into v, removing the original two-variable dependence and producing an equation in v and x.

3. What equation family does this calculator solve?

This calculator handles the common homogeneous family dy/dx = k(y/x)^n. It evaluates the closed-form solution, computes slope values, creates a graph, and builds an exportable result table.

4. What happens when n equals 1?

The equation becomes dy/dx = k(y/x). After substitution, the solution reduces to the power-law form y = Cx^k. The calculator switches to that special-case formula automatically.

5. Why do some inputs return an invalid result?

Certain constants can produce non-real intermediate values, especially when a negative base is raised to a fractional power. In that case, the calculator asks for different values to keep the solution real.

6. What does the integration constant C mean?

C selects one specific member from the full family of solutions. Changing C shifts the curve while preserving the same differential equation structure.

7. What does the plotted curve represent?

It shows the computed function y(x) over your chosen interval. The table also reports v = y/x and dy/dx at each sampled point, which helps verify the equation numerically.

8. Can I use this tool for learning and checking homework?

Yes. It is useful for checking algebra, confirming graphs, and understanding the substitution method. You should still show complete derivations when your instructor requires full manual working.