Circuit Theory-2 #14 | Laplace Transform Properties - Shift, Differentiation, Integration

Instructor: Dr. Süleyman Burak ÇELİK

In this Circuit Theory-2 lesson we add the manipulation tools to the Laplace transform: time shift, frequency shift, differentiation, integration, and multiplication by t. With these in hand, every signal in the table can be transformed by inspection — and ODEs collapse into algebra. You will learn: - Time-shift theorem: f(t-a) u(t-a) ↔ e^{-as} F(s) - Frequency-shift (s-shift): e^{-at} f(t) ↔ F(s+a) - Differentiation in time: f'(t) ↔ s F(s) - f(0) - Integration in time: ∫₀^t f(τ) dτ ↔ F(s) / s - Multiplication by t: t · f(t) ↔ -F'(s) - A 7-line property table for quick reference - Worked example: y' + 2y = 0, y(0) = 1 — solved by pure algebra in s Timestamps: 00:00 Properties of the Laplace Transform 00:03 From pairs to manipulation rules 00:32 Time shift theorem 01:25 Frequency shift theorem 02:11 Differentiation in time 03:12 Integration in time 03:56 Bonus: multiplication by t 04:24 Properties summary table 05:11 Worked example: ODE → algebra 06:12 Wrap Next lesson: inverse Laplace and partial fractions. Part of the Circuit Theory-2 playlist by AcEdumy.