Calculator inputs
Use initial conditions to solve C1 and C2. Otherwise, manual constants generate a chosen particular curve.
Formula used
The calculator solves the homogeneous Cauchy Euler differential equation a x² y″ + b x y′ + c y = 0. Assume a trial solution y = xm. Substitution gives the indicial equation:
Then the solution follows these rules:
- Distinct real roots m1, m2: y = C1xm1 + C2xm2
- Repeated root m: y = (C1 + C2ln x)xm
- Complex roots α ± iβ: y = xα[C1cos(β ln x) + C2sin(β ln x)]
When initial conditions are supplied, the page solves a 2×2 linear system for C1 and C2, then evaluates y(x) and y′(x) over the chosen domain.
How to use this calculator
- Enter coefficients a, b, and c from the equation form a x² y″ + b x y′ + c y = 0.
- Enter x0, y(x0), and y′(x0) to solve the integration constants automatically.
- Leave initial conditions blank if you prefer manual C1 and C2 values.
- Choose an evaluation point x, plus a graph start, graph end, and number of points.
- Press Solve equation to view the symbolic family, fitted constants, numerical evaluation, Plotly graph, and downloadable table.
Example data table
This sample shows the default equation x²y″ + 2xy′ − 2y = 0 with y(1)=2 and y′(1)=1.
| Parameter | Example value | Meaning |
|---|---|---|
| a | 1 | Coefficient of x²y″ |
| b | 2 | Coefficient of xy′ |
| c | -2 | Coefficient of y |
| x0 | 1 | Initial-condition point |
| y(1) | 2 | Function value at x0 |
| y′(1) | 1 | Derivative value at x0 |
| Root type | Distinct real roots | m = 1 and m = -2 |
| Particular solution | y = x + x-2 | Computed from the initial conditions |
Frequently asked questions
1. What kind of differential equation does this solver handle?
It handles second-order homogeneous Cauchy Euler equations of the form a x² y″ + b x y′ + c y = 0. The tool classifies roots, builds the general solution, and can fit constants from initial conditions.
2. Why must x stay positive in this page?
The repeated-root and complex-root solutions use ln(x). To keep the formulas consistent and real-valued for ordinary use, the calculator restricts evaluation to positive x values only.
3. What happens when the indicial roots are equal?
A repeated indicial root creates the logarithmic second term. The solver returns y = (C1 + C2 ln x)x^m and still solves the constants from the provided initial conditions.
4. How are complex roots displayed?
When the discriminant is negative, the roots are written as α ± iβ. The calculator converts that form into the real-valued solution x^α[C1 cos(β ln x) + C2 sin(β ln x)].
5. Can I use the tool without initial conditions?
Yes. Leave x0, y(x0), and y′(x0) blank. Then enter manual C1 and C2 values to graph a chosen member of the solution family, or let the defaults C1=1 and C2=0 run.
6. What does the transformed equation mean?
Using t = ln(x) converts the equidimensional equation into a constant-coefficient equation in u(t). That explains why the roots of the indicial equation fully determine the solution structure.
7. What is included in the CSV and PDF exports?
The exports include the computed solution table built from your graph range. The PDF also adds a short summary containing the root type, indicial equation, constants, and evaluated output values.
8. Does the calculator solve nonhomogeneous forcing terms?
No. This page focuses on the standard homogeneous second-order form. For forcing terms on the right side, you would need a separate method, such as variation of parameters or a tailored trial solution.