Calculator
Formula Used
For the homogeneous equation:
ay'' + by' + cy = 0
The characteristic equation is:
ar² + br + c = 0
The repeated root condition is:
b² - 4ac = 0
The repeated root is:
r = -b / 2a
The solution is:
y = (C1 + C2x)erx
The derivative is:
y' = [C2 + r(C1 + C2x)]erx
How to Use This Calculator
- Enter coefficients a, b, and c from ay'' + by' + cy = 0.
- Use values that make b² - 4ac equal to zero.
- Enter C1 and C2 if constants are already known.
- Check initial condition mode to solve constants automatically.
- Enter an x value for direct evaluation.
- Set the table start, end, and step values.
- Press Calculate to show results below the header.
- Use CSV or PDF buttons to save the output.
Example Data Table
| a | b | c | Discriminant | Repeated Root | Example Solution |
|---|---|---|---|---|---|
| 1 | 4 | 4 | 0 | -2 | y = (2 + x)e-2x |
| 2 | -8 | 8 | 0 | 2 | y = (3 - x)e2x |
| 4 | 12 | 9 | 0 | -1.5 | y = (1 + 4x)e-1.5x |
Understanding Repeated Root Equations
A second order differential equation with constant coefficients often appears in vibration, circuits, controls, and growth models. The repeated root case is special because the characteristic equation has only one root. That happens when the discriminant equals zero. Instead of two different exponential terms, the solution needs an extra x multiplier. This calculator focuses on that exact case, so the output stays direct, clear, and useful for study.
Why the Repeated Root Matters
When roots repeat, two normal exponential solutions would be identical. A differential equation still needs two independent solution parts. The term x erx supplies the missing independent part. This method is common in courses covering homogeneous linear differential equations. It also helps when checking damping models, where critical damping creates a repeated root.
What the Calculator Checks
The tool accepts coefficients a, b, and c from a y'' + b y' + c y = 0. It computes the discriminant, root, general solution, optional constants, derivative, and selected function value. If you provide initial conditions, it solves C1 and C2 automatically. If you do not provide them, it uses your entered constants for evaluation and the table.
Practical Study Benefits
Students can compare symbolic work with numeric output. Teachers can prepare examples faster. Engineers can test a critical damping model before adding units or forcing terms. The table also shows how the solution changes across several x values. That makes the exponential trend easier to inspect.
Reading the Result
Start with the discriminant line. If it is not zero, the equation does not belong to the repeated root case. Next read r, then the solution form. When initial conditions are active, the solved constants appear below the formula. Finally, use the downloadable files to save the calculation for assignments, reports, or classroom notes. Always round only at the final step when accuracy matters.
Common Mistakes to Avoid
Do not confuse repeated roots with close roots. A very small discriminant may come from rounding, but the exact equation decides the method. Also remember that coefficient a cannot be zero. If a is zero, the equation is no longer second order. Use consistent initial values and keep x0 in the same variable scale before trusting final results.
FAQs
What is a repeated root differential equation?
It is a second order equation whose characteristic equation has the same root twice. This happens when b² - 4ac equals zero.
What solution form is used for repeated roots?
The standard form is y = (C1 + C2x)erx. The x term keeps the two solution parts independent.
Can this calculator solve distinct roots?
No. It checks the discriminant and focuses on the repeated-root case. Distinct roots need a different solution form.
What does coefficient a represent?
Coefficient a multiplies y''. It cannot be zero because the equation would no longer be second order.
Can I use initial conditions?
Yes. Check the initial condition option, then enter x0, y(x0), and y'(x0). The calculator solves C1 and C2.
Why is the discriminant important?
The discriminant tells the root type. A zero discriminant confirms the repeated-root method is appropriate.
What does the table show?
The table shows x, y, and y' values across your selected range. It helps inspect the solution numerically.
Can I download the result?
Yes. After calculation, use the CSV or PDF button to save the result, formula values, and generated table.