Laplace Transform
Turn ODEs into algebra.
Key idea
Turn ODEs into algebra.
In depth
ℒ{f}(s) = ∫₀^∞ e^(−st)f(t) dt turns differentiation into multiplication by s and ODEs into algebra: ℒ{f′} = sF − f(0). Tables give common transforms, and the inverse recovers f.
Worked example
ℒ{e^(at)} = 1/(s − a), ℒ{sin t} = 1/(s² + 1).
Formula
Real-world uses
- Solving control system and circuit problems by turning calculus into algebra
Common mistake
Forgetting the initial-condition terms when transforming derivatives.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Laplace Transform?
Turn ODEs into algebra. It is a topic at the expert level in Differential Equations, Abstract Algebra & Applied Math.
What is the formula for Laplace Transform?
The key formula is L{f}(s) = ∫₀^∞ f(t)e⁻ˢᵗ dt.
Where is Laplace Transform used in real life?
Real-world examples: Solving control system and circuit problems by turning calculus into algebra.
What should I learn before Laplace Transform?
Make sure you are comfortable with RLC Circuits, the previous topic in Differential Equations, Abstract Algebra & Applied Math.
What is a common mistake with Laplace Transform?
Forgetting the initial-condition terms when transforming derivatives.
How can I practise Laplace Transform?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Laplace Transform?
Move on to Laplace Transform Method next.
Related expert topics in Differential Equations, Abstract Algebra & Applied Math
- Exact Equations – M dx + N dy = 0 with Mᵧ = Nₓ.
- Bernoulli Equations – Substitute v = y¹⁻ⁿ to linearise.
- Runge–Kutta Methods – Higher-accuracy numerical solvers.
- Second-Order Homogeneous Equations – Use the characteristic equation.
- Undetermined Coefficients – Guess the particular solution form.
- Variation of Parameters – General method for particular solutions.