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Laplace of derivatives

The Laplace transform turns a derivative into an algebraic factor $s$ plus an initial-condition term. This is why Laplace transforms differential equations into algebraic equations that are easy to solve.

\mathcal L\{f'(t)\} = sF(s) - f(0)

transform of the first derivative

\mathcal L\{f''(t)\} = s^2F(s) - sf(0) - f'(0)

transform of the second derivative

Symbols

F(s)

the Laplace transform of f(t)

Example

Solve $y'+3y=0$ , $y(0)=4$ : $sY-4+3Y=0 \Rightarrow Y(s)=\dfrac{4}{s+3}$ , so $y=4e^{-3t}$ .

Transform the whole equation, solve for

Y(s)

, then use a table backwards (inverse transform).

Practise the Laplace transform for free →

← The Laplace transform · Shift rule and partial fractions →

Part of Laplace, Fourier and PDEs: The Laplace transform.