Solve transform expressions from common linear models accurately. Inspect initial conditions, forcing choices, and poles. Download clean tables and plots for guided math practice.
| Example | Differential model | Initial values | Forcing | Computed Y(s) |
|---|---|---|---|---|
| 1 | y'' + 3y' + 2y = 5sin(4t) | y(0)=1, y'(0)=0 | F(s)=20/(s²+16) | [20/(s²+16) + s + 3] / (s²+3s+2) |
| 2 | 2y'' + 5y' + y = 3e^(2t) | y(0)=0, y'(0)=1 | F(s)=3/(s-2) | [3/(s-2) + 2] / (2s²+5s+1) |
For the model a·y''(t) + b·y'(t) + c·y(t) = f(t), take the Laplace transform of every term.
L{y'(t)} = sY(s) - y(0)
L{y''(t)} = s²Y(s) - sy(0) - y'(0)
After substitution, the transformed equation becomes:
a[s²Y(s) - sy(0) - y'(0)] + b[sY(s) - y(0)] + cY(s) = F(s)
Rearranging for Y(s):
Y(s) = [F(s) + a·s·y(0) + a·y'(0) + b·y(0)] / [a·s² + b·s + c]
Common forcing transforms used in this page are:
This calculator helps you form Y(s) from a linear differential equation with initial conditions. It is designed for classroom practice, homework checks, and quick review. You enter coefficients, select a forcing term, and the page builds the transformed solution step by step.
Many differential equations are easier to solve after transformation. The Laplace method changes derivatives into algebraic terms involving s. That makes rearrangement faster and clearer. Once Y(s) is known, you can continue toward partial fractions or inverse transformation with better structure.
The tool supports zero input, constants, exponentials, sine signals, cosine signals, and polynomial forcing. These cover many standard problems in maths and engineering courses. Initial values are built directly into the numerator, so the calculator shows how the starting state changes the transformed response.
Besides the symbolic form, the page also evaluates Y(s) on a real-axis interval that you choose. This is useful for spotting growth, decay, or undefined points near poles and singular terms. The table helps learners compare values instead of relying only on symbolic algebra.
The Plotly graph adds a visual layer. You can see how the transformed function behaves as s changes. Sudden spikes often indicate poles or nearby singularities. Smooth decay often suggests stronger denominator growth. This visual check can make transform work easier to interpret during revision.
CSV export is useful for spreadsheets and saved classwork. PDF export is useful for printing or sharing a clean result sheet. Both options help you keep a copy of the sampled values, especially when comparing several differential models side by side.
It forms Y(s) for equations of the type a·y''(t) + b·y'(t) + c·y(t) = f(t). It uses initial values and a selected forcing transform.
Yes. Choose the zero option when the equation is homogeneous. The calculator will then use only the initial-condition terms in the numerator.
Undefined values usually happen at poles or forcing singularities. Examples include s = 0 for constants or s = k for exponential forcing.
No. This page focuses on building and sampling Y(s). It is best used before inverse Laplace steps such as factorization or partial fractions.
Poles are the roots of the denominator. They help describe important system behavior and indicate where Y(s) becomes unbounded or sensitive.
Derivative transforms generate extra terms containing y(0) and y'(0). Those terms move into the numerator when the equation is rearranged for Y(s).
Yes. Use the CSV button for spreadsheet-friendly data. Use the PDF button for a printable summary and sampled value table.
Select Polynomial. Then enter the coefficient A and the integer power n. The page uses A·n!/s^(n+1) for the transform.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.