01
Forward transforms
Convert common time-domain functions into the s-domain using linearity and transform rules.
Laplace Transform Solver
Purpose-built solver page
Laplace Transform Solver
Use this Laplace transform solver for worked help with forward or inverse transforms and common initial-value setups. Include the time/frequency variable and initial conditions explicitly.
Type or upload a problemWorked explanationsNo sign-up required
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ℒ{f(t)} = F(s)
Forward transforms
Inverse transforms
Shift properties
laplace transform solver
Solve your laplace transform problem
Enter your laplace transform problem below. Add the variable, conditions, and method you want so the AI can return a more relevant worked response.
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laplace transform solver
What this Laplace Transform solver is designed for
The page is organized around the problem types people actually search for, while staying within the text-and-image solver's current capabilities.
02
Inverse transforms
Recognize table forms and use algebra or partial fractions before transforming back.
03
Shift properties
Apply exponential and time-shift rules while tracking notation and conditions.
04
Initial-value problems
Transform derivatives with their initial values, solve algebraically for the transform, then invert.
1
ℒ{1} = 1/stⁿ
ℒ{tⁿ} = n!/sⁿ⁺¹eᵃᵗ
ℒ{eᵃᵗ} = 1/(s-a)sin(bt)
ℒ{sin(bt)} = b/(s²+b²)cos(bt)
ℒ{cos(bt)} = s/(s²+b²)Step
How to use the Laplace Transform solver
A complete prompt usually produces a more useful explanation than an isolated expression.
- 1
State the direction
f(t) ⇄ F(s)Say whether you need a Laplace transform or an inverse Laplace transform.
- 2
Define variables and conditions
y(0), y′(0), …Use t for time, s for the transform variable, and include initial values or step functions.
- 3
Simplify into known forms
ℒ{y′} = sY(s) - y(0)Use linearity, shifts, or partial fractions to match transform-table entries.
- 4
Transform back and check
Y(s) → y(t)For an IVP, verify the resulting function against the differential equation and initial conditions.
Tip: ask for a named method, a domain, or a verification step when your course requires one.
Example
Laplace Transform problems to try
These examples reflect common search intent and can be inserted into the solver as a starting point.
Example 1
Find L{t^2 e^(-3t)}A forward transform combining a power of t with an exponential shift.
Example 2
Find L^-1{(2s + 5)/(s^2 + 4s + 13)}An inverse transform prepared by completing the square.
Example 3
Solve y' + 2y = 4, y(0) = 1 using Laplace transformsAn initial-value problem where the derivative transform carries the initial condition.
y′ + ay = g(t)↓ ℒ(s+a)Y(s) = G(s) + y(0)↓ ℒ⁻¹Y(s) → y(t)Common mistakes to watch for
A worked answer is most useful when you know which details are easy to lose along the way.
Omitting initial conditions
Transforms of derivatives depend on the supplied initial values.
Confusing t and s
Keep the time-domain and transform-domain variables distinct.
Skipping algebra before inversion
Completing the square or partial-fraction decomposition is often required to match a table form.
Laplace Transform solver FAQ
