Computing L{t²e⁻³ᵗ}

Differential Equations · Axiom Academy

Use the Laplace transform definition and integration by parts to find Find the Laplace transform of f(t) = t^2e^ -3t directly from the definition , using integration by parts. Excellent work! You've computed from first principles using integration by parts. Here's what we learned: Definition first: Always start with the integral definition — never jump straight to a table. Integration by parts: for a polynomial times an exponential, apply integration by parts repeatedly. A degree- n polynomial needs n rounds of integration by parts. Boundary behavior: for s > -3 , both t^2e^ -(s+3)t and te^ -(s+3)t vanish as — this is what makes the boundary terms drop to 0 . Pattern recognition: the result matches the general rule . We can check our result against the standard Laplace transform formulas: , where (the first-shifting theorem) Shifting with a = -3 : — matching our integration-by-parts answer exactly. Working this from first principles builds the intuition behind the transform tables — even though the shifting theorem gets you there in one line.

This is the written version of the interactive lesson above. See the full Differential Equations course.