Calculator Form
Formula Used
Start with the differential equation y'' + py' + qy = 0.
The characteristic equation is r² + pr + q = 0.
Complex eigenvalues occur when Δ = p² - 4q < 0.
Then r = α ± βi, where α = -p/2 and β = √(4q - p²)/2.
The real general solution is y(x) = e^(αx)[C1 cos(βx) + C2 sin(βx)].
If initial conditions are known, then C1 = y(0) and C2 = [y'(0) - αy(0)] / β.
How to Use This Calculator
- Select coefficient mode or direct root mode.
- Enter p and q, or enter α and β.
- Make sure the case represents complex eigenvalues.
- Add x if you want function evaluation at a point.
- Enter C1 and C2 for a specific solution.
- Or enter y(0) and y'(0) to compute constants automatically.
- Press the button to view the result above the form.
- Use the CSV or PDF buttons to export the table.
Example Data Table
| p | q | α | β | Eigenvalues | General Solution |
|---|---|---|---|---|---|
| 2 | 5 | -1 | 2 | -1 ± 2i | e^(-x)[C1 cos(2x) + C2 sin(2x)] |
| -4 | 13 | 2 | 3 | 2 ± 3i | e^(2x)[C1 cos(3x) + C2 sin(3x)] |
| 1 | 10 | -0.5 | 3.1225 | -0.5 ± 3.1225i | e^(-0.5x)[C1 cos(3.1225x) + C2 sin(3.1225x)] |
Complex Eigenvalues and Real Solutions
Why this topic matters
Complex eigenvalues appear in many second-order differential equations. They show that a solution oscillates. They also show whether that oscillation grows, stays steady, or decays. This calculator turns that theory into a quick working result.
What the calculator finds
The tool starts from the equation y'' + py' + qy = 0. It also accepts direct values for α and β. From either path, it builds the complex roots and the matching real general solution. That saves time during homework, revision, and checking.
How the solution is interpreted
When the roots are α ± βi, the factor e^(αx) controls amplitude. The cosine and sine terms control oscillation. A negative α means decay. A positive α means growth. A zero α means a steady oscillation with constant amplitude.
Useful outputs for study
This page shows the discriminant, eigenvalues, characteristic equation, basis solutions, oscillation period, and frequency. It also evaluates y(x) and y'(x) when constants are available. That makes it useful for classroom examples and worked solutions.
Using initial conditions
Many problems give y(0) and y'(0). This calculator uses those values to solve for C1 and C2 automatically. That helps you move from the general solution to a specific answer without doing extra algebra by hand.
Best situations for this tool
Use it for characteristic equations with negative discriminants, mechanical vibration models, signal response problems, and differential equations practice. The export buttons also help when you need a neat record of results for notes, assignments, or quick comparison.
FAQs
1. What does this calculator solve?
It solves second-order linear differential equations that produce complex eigenvalues. It returns the real general solution, related basis functions, and optional evaluated values when constants or initial conditions are given.
2. When do complex eigenvalues occur?
They occur when the discriminant p² - 4q is negative. That means the characteristic equation has no distinct real roots and instead produces a conjugate pair α ± βi.
3. Why is the final answer real?
The conjugate roots combine into cosine and sine terms. Those terms are real-valued, so the calculator rewrites the solution in a real form that is easier to use and interpret.
4. What does α represent?
α is the real part of the eigenvalues. It controls the envelope e^(αx). Negative α causes decay, positive α causes growth, and zero α gives steady oscillation.
5. What does β represent?
β is the imaginary part magnitude. It controls oscillation speed. A larger β means more rapid oscillation and a shorter period, since the period equals 2π divided by β.
6. Can I use initial conditions instead of C1 and C2?
Yes. Enter y(0) and y'(0). The calculator then computes C1 and C2 automatically. Those values override any manually entered constants for the specific solution.
7. What if my coefficients give real roots?
This tool is built for the complex-root case only. If the discriminant is zero or positive, you need the repeated-root or real-root form instead of the sine-cosine solution.
8. Why export to CSV or PDF?
Exports help you save a clean result table. They are useful for assignments, revision notes, tutoring sessions, and comparing several differential equation cases side by side.