Laplace transform general form

Calculus Level 3

For a second order linear ODE of the form $ay''+by'+cy=f(t)$ , where $a,b,c$ are constants, which of these is the correct expression for $\mathcal{L}[y]=Y(s)$ ?

\frac{L[f(t)]-(as+b)y(0)-ay'(0)}{as^2+bs+c}

\frac{(as+b)y(0)+ay'(0)}{a^2s+bs+c}

\frac{L[f(t)]+(as+b)y(0)+ay'(0)}{a^2s+bs+c}

\frac{L[f(t)]+(as+b)y(0)+ay'(0)}{as^2+bs+c}

1 solution

Karan Chatrath
Mar 19, 2020

Consider the differential equation:

$a \ddot{y} + b \dot{y} +cy = f(t)$

Taking Laplace transform on both sides gives:

$a \left(s^2 Y(s) -s y(0) -\dot{y}(0)\right) + b \left(sY(s) - y(0)\right) + cY(s) = \mathscr{L}\left[f(t)\right]$

Where: $Y(s) = \mathscr{L}\left[y(t)\right]$

Re-arranging gives:

$Y(s) = \frac{\mathscr{L}\left[f(t)\right] + \left(as+b\right)y(0) + a\dot{y}(0)}{as^2+bs+c}$

Use '\mathcal{L}' or '\mathscr{L}' to get the curvy L symbol. The two are slightly different but for me, either symbol looks okay.

Laplace transform general form

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