• Recall:

  • $\lambda ,e^{\lambda x}$ are eigenvalues, eigenfunctions of the differentiation operator $D=\frac{d}{dt}$

    ${\displaystyle D{e}^{\lambda t}=\lambda e^{\lambda t}}$

  • Constant coefficient ODEs

    ${\displaystyle D^{n}y+a_{n-1}{D}^{n-1}y+\cdots +a_{1}Dy+a_{0}y=0}$

    have a general solution

    ${\displaystyle y\left(t\right)=c_1{e}^{r_{1}t}+\cdots +c_n{e}^{r_{n}t},}$

    a linear combination of eigenfunctions of $D$ with $r_1,\dots ,r_n$ solutions of the algebraic, characteristic eq.

    ${\displaystyle r^n+a_{n-1}{r}^{n-1}+\cdots +a_{1}r+a_0=0.}$

• Idea: seek a more general linear combination of eigenfunctions $e^{-st}$ with a continuous range of eigenvalues

  • denote the coefficients of the linear combination as $f\left(t\right)$

  • choose $s⩾0$, and $e^{-st}$ to avoid exponentially increasing functions

  • replace summation of $e^{r_{i}t}$ for a finite linear combination with integration over $s$ values

Definition. The Laplace transform of $f:[0,\infty )\to ℝ$ is

and $f\left(t\right)={ℒ}^{-1}\left(F\right)\left(t\right)$, $f={ℒ}^{-1}\left(F\right)$ is the inverse Laplace transform.