Enter Calculator Inputs
Enter forcing coefficients as comma-separated values from highest power to constant.
Example: 1, 0 means s.
Example: 1 over 1, 0 means 1/s.
Example Data Table
| Equation | Initial Values | F(s) | Y(s) |
|---|---|---|---|
| y'' + 5y' + 6y = 1 | y(0)=1, y'(0)=0 | 1 / s | (s^2 + 5s + 1) / (s^3 + 5s^2 + 6s) |
| 2y'' + 3y' + y = e^(-t) | y(0)=0, y'(0)=2 | 1 / (s + 1) | (4s + 5) / ((s + 1)(2s^2 + 3s + 1)) |
| y'' + 4y = sin(2t) | y(0)=0, y'(0)=1 | 2 / (s^2 + 4) | (s^2 + 6) / (s^2 + 4)^2 |
Formula Used
Model equation
a2y''(t) + a1y'(t) + a0y(t) = f(t)
Laplace transform step
a2[s²Y(s) - sy(0) - y'(0)] + a1[sY(s) - y(0)] + a0Y(s) = F(s)
Direct Y(s) form
Y(s) = [F(s) + a2sy(0) + a2y'(0) + a1y(0)] / [a2s² + a1s + a0]
Rational forcing form
If F(s) = Nf(s)/Df(s), then Y(s) = [Nf(s) + (a2sy0 + a2y1 + a1y0)Df(s)] / [Df(s)(a2s² + a1s + a0)]
The calculator multiplies and adds the required polynomials automatically. It then plots the computed transform across positive real-axis values of s.
How to Use This Calculator
- Enter the second-order equation coefficients a2, a1, and a0.
- Enter the initial values y(0) and y'(0).
- Describe the forcing transform using comma-separated polynomial coefficients.
- Use highest-power-first order for both numerator and denominator entries.
- Click Calculate Y(s) to generate the symbolic result.
- Review the displayed numerator, denominator, degree classification, and sample value.
- Study the graph and computed sample table.
- Download the result as CSV or PDF when needed.
FAQs
1. What does this calculator find?
It computes the Laplace-domain output Y(s) for a second-order linear differential equation with initial conditions and a rational forcing transform F(s).
2. How should I enter polynomial coefficients?
Enter coefficients from highest power to constant, separated by commas. For example, s² + 3s + 2 becomes 1, 3, 2.
3. What does 1 over 1, 0 represent?
That represents F(s) = 1/s. It is the transform of a unit step input in many standard Laplace problems.
4. Why must a2 be non-zero?
This page is designed for second-order equations. If a2 were zero, the model would reduce to a lower-order problem and require different handling.
5. What does the plotted curve mean?
The graph shows Y(s) evaluated along positive real-axis values. It helps you inspect growth, decay, asymptotes, and possible singular behavior visually.
6. What does strictly proper mean?
A rational function is strictly proper when its numerator degree is smaller than its denominator degree. This is common in stable transfer forms.
7. Can this calculator simplify factors?
It expands and combines the required polynomials cleanly. It does not perform symbolic factor cancellation or partial fraction decomposition.
8. When should I export CSV or PDF?
Use CSV for data work, spreadsheets, and plotting elsewhere. Use PDF for reporting, printing, or sharing a concise transform summary.