Using First Shifting Theorem

Differential Equations · Axiom Academy

EXAMPLE Using the First Shifting Theorem Find using the exponential-shift property Find , given the known transform . Excellent work! You've applied the First Shifting Theorem from start to finish. Here's what to remember: Pattern recognition: whenever you see e^ at f(t) , reach for the First Shifting Theorem. The transformation: if , then — replace every s in F(s) with s - a . Sign awareness: for e^ -2t , we have a = -2 , so s - a becomes s - (-2) = s + 2 . The double negative is the step students miss most. Algebraic care: after substituting, expand squared terms carefully — (s+2)^2 = s^2 + 4s + 4 , not s^2 + 4 or s^2 + 2s + 4 . Verification: the final answer is a rational function of s with no exponentials remaining. The First Shifting Theorem lets you handle exponentially damped or growing functions by leveraging transforms you already know. Practice it with different functions to build your intuition for the sign of a .

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