Laplace Transform & ODE Solver
Built for engineering and BS Mathematics students. Find Laplace and inverse Laplace transforms, or solve a linear differential equation with initial conditions, with every step shown and the final answer checked numerically.
Use * for multiplication (2t and 3sin(2t) also work), ^ for powers, e^(...) or exp(), sin, cos, sinh, cosh. In ODEs write y', y'', y''' (up to third order, constant coefficients).
What it can solve
| Mode | Handles |
|---|---|
| Laplace | Sums of tn, eat, sin, cos, sinh, cosh and their products, using the table, first shifting theorem and multiplication by tn |
| Inverse Laplace | Proper rational functions N(s)/D(s): real, repeated and complex roots via partial fractions |
| ODE | Linear equations with constant coefficients up to order 3, any forcing term above, with initial conditions |
Worked example
Solve y′′ + 3y′ + 2y = 0 with y(0) = 1, y′(0) = 0. Transform: (s²Y − s) + 3(sY − 1) + 2Y = 0, hence Y = (s + 3)/((s+1)(s+2)) = 2/(s+1) − 1/(s+2). Inverse: y = 2e−t − e−2t.
How the answer is checked
For equations the solution is differentiated and substituted back into the ODE at several values of t, and the initial conditions are tested. For inverse transforms the partial fractions are recombined and compared with F(s). A green “verified” badge means those tests passed.
Not supported yet
Unit step and Dirac delta functions, variable-coefficient equations, systems of equations, and non-linear equations. In those cases Calvo shows a message instead of guessing.
FAQ
How do I solve a differential equation with the Laplace transform?
Transform both sides using L{y'} = sY − y(0) and L{y''} = s²Y − s y(0) − y'(0). Solve the algebraic equation for Y(s), split into partial fractions and take the inverse transform.
What is the Laplace transform of sin(at)?
a / (s² + a²).
What is the first shifting theorem?
If L{f(t)} = F(s) then L{eat f(t)} = F(s − a).