Solve Cauchy-Euler equations with roots and constants quickly. Inspect worked steps and function behavior clearly. Export results for class notes, practice sets, and revision.
Use this solver for second-order Cauchy-Euler equations of the form ax²y″ + bxy′ + cy = 0 with x > 0.
For the Euler equation ax²y″ + bxy′ + cy = 0, assume a trial solution y = xm. Then:
y′ = mxm-1
y″ = m(m-1)xm-2
Indicial equation: am² + (b-a)m + c = 0
The root pattern determines the solution form:
If initial conditions are given, the solver forms a 2×2 linear system and computes C₁ and C₂ numerically.
It solves homogeneous second-order Cauchy-Euler equations written as ax²y″ + bxy′ + cy = 0, where x stays positive for logarithmic terms.
Repeated and complex-root solutions use ln x. Real logarithms require x > 0, so the calculator keeps input points in that valid domain.
The discriminant of the indicial equation reveals whether the roots are distinct real, repeated real, or complex. That directly selects the correct solution form.
Yes. Provide x₀, y(x₀), and y′(x₀), then enable initial-condition solving. The calculator builds a linear system and computes both constants.
Yes, after constants are known. Enter a positive evaluation point, enable numeric evaluation, and the page returns both y(x) and y′(x).
Then the equation is no longer second-order Euler form. The solver blocks that case because the characteristic setup would not apply correctly.
Yes. The CSV and PDF exports include the main result fields and the step summary so you can keep a compact worked record.
| a | b | c | x₀ | y(x₀) | y′(x₀) | x |
|---|---|---|---|---|---|---|
| 1 | 1 | -1 | 1 | 2 | 1 | 2 |
| 1 | 3 | 1 | 1 | 4 | 2 | 3 |
| 2 | 2 | 5 | 1 | 1 | 0 | 2 |
The first row is loaded by the example button and demonstrates distinct real roots with solvable constants.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.