Calculator Inputs
Use the responsive grid below. It shows three columns on large screens, two on medium screens, and one on mobile devices.
Example Data Table
This example uses x²y″ + 3xy′ + 2y = 0 with y(1) = 3 and y′(1) = -4, giving y(x) = 2x-1 + x-2.
| x | Equation | Roots | Specific Solution | y(x) |
|---|---|---|---|---|
| 1 | x²y″ + 3xy′ + 2y = 0 | m₁ = -1, m₂ = -2 | 2x-1 + x-2 | 3 |
| 2 | x²y″ + 3xy′ + 2y = 0 | m₁ = -1, m₂ = -2 | 2x-1 + x-2 | 1.25 |
| 4 | x²y″ + 3xy′ + 2y = 0 | m₁ = -1, m₂ = -2 | 2x-1 + x-2 | 0.5625 |
Formula Used
Standard equation: a x2y″ + bxy′ + cy = 0
Trial form: y = |x|m
Characteristic equation: a m(m - 1) + bm + c = 0, which simplifies to a m2 + (b - a)m + c = 0.
Distinct real roots: y = C1|x|m₁ + C2|x|m₂
Repeated root m: y = (C1 + C2 ln|x|)|x|m
Complex roots α ± iβ: y = |x|α[C1 cos(β ln|x|) + C2 sin(β ln|x|)]
How to Use This Calculator
- Enter coefficients a, b, and c for the homogeneous Cauchy Euler equation.
- Select whether your working interval lies to the right or left of zero.
- Choose a plotting interval that stays entirely on the same side of zero.
- Keep initial conditions enabled if you want exact constants C1 and C2.
- Enter x₀, y(x₀), and y′(x₀) when solving for a unique specific solution.
- Click Solve Equation to display the result above the form.
- Use the CSV button for numerical output and the PDF button for a shareable report.
FAQs
1) What kind of differential equation does this page solve?
It solves second-order homogeneous Cauchy Euler equations of the form a x²y″ + bxy′ + cy = 0. It does not handle forcing terms or higher-order systems.
2) Why must the interval avoid x = 0?
The point x = 0 is singular for a Cauchy Euler equation. The theory and formulas apply on intervals entirely within x > 0 or entirely within x < 0.
3) What happens when the discriminant is positive?
A positive discriminant gives two distinct real roots. The solution becomes a linear combination of two power functions, each built from one root.
4) What happens for a repeated root?
A repeated root produces one pure power solution and one logarithmic companion. That is why the result includes a term containing ln|x|.
5) Why do complex roots create sine and cosine terms?
Complex roots lead to oscillation in the logarithmic variable ln|x|. The real-valued solution is rewritten with cosine and sine to keep the answer practical.
6) What do the initial conditions change?
Initial conditions determine C₁ and C₂, converting the general family into one specific solution. Without them, the graph uses a representative choice for the constants.
7) Can I use negative x-values?
Yes. Choose the x < 0 domain option and keep x₀, plot start, and plot end all negative. The solver uses |x| in the closed-form expressions.
8) What does the CSV file contain?
The CSV export contains graph sample points with x and y(x) values. It is useful for spreadsheet checks, reports, and quick verification outside the browser.