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Model a second-order linear differential equation using Laplace-ready coefficients, initial values, and two selectable forcing terms.
Formula used
This calculator models the second-order linear equation a y″ + b y′ + c y = f(t) with initial conditions y(0) and y′(0).
Applying Laplace transforms gives:
a[s²Y(s) - s y(0) - y′(0)] + b[sY(s) - y(0)] + cY(s) = F(s)
Rearranging gives the working solver form:
Y(s) = [F(s) + a s y(0) + a y′(0) + b y(0)] / [a s² + b s + c]
The calculator also inspects the characteristic equation a r² + b r + c = 0 to classify roots, damping behavior, and stability.
How to use this calculator
- Enter the coefficients for y″, y′, and y.
- Provide the initial position and initial slope.
- Set the evaluation time and the response table end time.
- Choose one or two forcing terms, then fill only the fields those terms need.
- Press Solve ODE to display the result above the form.
- Use the CSV and PDF buttons to save the response summary and table.
Example data table
Example case: y″ + 5y′ + 6y = 10, y(0) = 1, y′(0) = 0.
| t | f(t) | y(t) | y′(t) |
|---|---|---|---|
| 0 | 10 | 1.000000 | 0.000000 |
| 0.5 | 10 | 1.228415 | 0.578997 |
| 1 | 10 | 1.462379 | 0.342193 |
| 2 | 10 | 1.633340 | 0.063348 |
| 3 | 10 | 1.661874 | 0.009421 |
Frequently asked questions
1. What kind of equations does this solver handle?
It handles second-order linear ordinary differential equations with constant coefficients, initial conditions, and selectable forcing terms that match common Laplace-transform setups used in mathematics and engineering.
2. Why does the coefficient a need to be nonzero?
A zero value removes the second derivative term, so the equation stops being second-order. This page is designed specifically for second-order Laplace-domain models.
3. Does the solver show the Laplace-domain expression?
Yes. After submission, it prints the modeled equation, the transformed Y(s) expression, characteristic roots, damping information, and a sampled time-response table.
4. Can I combine two forcing inputs?
Yes. The calculator lets you add two forcing components, such as a constant with a sine input, or a delayed step with an exponential term.
5. Is the displayed result symbolic or numeric?
The page presents the Laplace-domain structure symbolically, then computes the time response numerically for the requested evaluation time and response table.
6. What do the roots tell me?
Real roots suggest non-oscillatory motion. Repeated roots mark critical behavior. Complex roots indicate oscillation, while the real part determines decay, growth, or stability.
7. When is the damping ratio shown?
It appears when the coefficients support a standard second-order form with positive a and c. Otherwise, the page leaves that metric unavailable.
8. What do the export buttons save?
The CSV file stores the response table. The PDF file includes the headline metrics, the modeled equation, the Laplace expression, and the sampled table.