CME 102 - Ordinary Differential Equations for Engineers


Calculus refresher
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Integral calculus

Primitive function ― The primitive function of a function $f$, noted $F$ and also known as an antiderivative, is a differentiable function such that:

\[\boxed{F'=f}\]

Integral ― Given a function $f$ and an interval $[a,b]$, the integral of $f$ over $[a,b]$, noted $\int_a^bf(x)dx$, is the signed area of the region in the $xy$-plane that is bounded by the graph of $f$, the $x$-axis and the vertical lines $x=a$ and $x=b$, and can be computed with the primitive of $f$ as follows:

\[\boxed{\int_a^bf(x)dx=F(b)-F(a)}\]

Integration by parts ― Given two functions $f, g$ on the interval $[a,b]$, we can integrate by parts the quantity $\int_a^bf(x)g'(x)dx$ as follows:

\[\boxed{\int_a^bf(x)g'(x)dx=\big[f(x)g(x)\big]_a^b-\int_a^bf'(x)g(x)dx}\]

Rational primitive functions ― The table below sums up the main rational functions associated to their primitives. We will omit the additive constant $C$ associated to all those primitives.

Function $f$ Primitive $F$
$\displaystyle a$ $\displaystyle ax$
$\displaystyle x^a$ $\displaystyle \frac{x^{a+1}}{a+1}$
$\displaystyle \frac{1}{x}$ $\displaystyle \ln|x|$
$\displaystyle \frac{1}{1+x^2}$ $\displaystyle \textrm{arctan}(x)$
$\displaystyle \frac{1}{1-x^2}$ $\displaystyle \frac{1}{2}\ln\left|\frac{x+1}{x-1}\right|$

Irrational primitive functions ― The table below sums up the main rational functions associated to their primitives. We will omit the additive constant $C$ associated to all those primitives.

Function $f$ Primitive $F$
$\displaystyle \frac{1}{\sqrt{1-x^2}}$ $\displaystyle \textrm{arcsin}(x)$
$\displaystyle -\frac{1}{\sqrt{1-x^2}}$ $\displaystyle \textrm{arccos}(x)$
$\displaystyle \frac{x}{\sqrt{x^2-1}}$ $\displaystyle \sqrt{x^2-1}$

Exponential primitive functions ― The table below sums up the main exponential functions associated to their primitives. We will omit the additive constant $C$ associated to all those primitives.

Function $f$ Primitive $F$
$\displaystyle \ln(x)$ $\displaystyle x\ln(x)-x$
$\displaystyle \exp(x)$ $\displaystyle \exp(x)$

Trigonometric primitive functions ― The table below sums up the main trigonometric functions associated to their primitives. We will omit the additive constant $C$ associated to all those primitives.

Function $f$ Primitive $F$
$\displaystyle \cos(x)$ $\displaystyle \sin(x)$
$\displaystyle \sin(x)$ $\displaystyle -\cos(x)$
$\displaystyle \tan(x)$ $\displaystyle -\ln|\cos(x)|$
$\displaystyle \frac{1}{\cos(x)}$ $\displaystyle \ln\left|\tan\left(\frac{x}{2}+\frac{\pi}{4}\right)\right|$
$\displaystyle \frac{1}{\sin(x)}$ $\displaystyle \ln\left|\tan\left(\frac{x}{2}\right)\right|$
$\displaystyle \frac{1}{\tan(x)}$ $\displaystyle \ln|\sin(x)|$

Laplace transforms

Definition ― The Laplace transform of a given function $f$ defined for all $t\geqslant0$ is noted $\mathscr{L}(f)$, and is defined as: \[\boxed{\mathscr{L}(f)=F(s)=\int_0^{+\infty}e^{-st}f(t)dt}\]

Remark: we note that $f(t)=\mathscr{L}^{-1}(F)$ where $\mathscr{L}^{-1}$ is the inverse Laplace transform.


Main properties ― The table below sums up the main properties of the Laplace transform:

Property $t$-domain $s$-domain
Linearity $\alpha f(t)+\beta g(t)$ $\alpha F(s)+\beta G(s)$
$t$-domain integral $\displaystyle\int_0^tf(\tau)d\tau$ $\displaystyle\frac{F(s)}{s}$
$t$-domain first derivative $f'(t)$ $sF(s)-f(0)$
$t$-domain second derivative $f''(t)$ $s^2F(s)-sf(0)-f'(0)$
$t$-domain $n^{th}$ derivative $f^{(n)}(t)$ $s^nF(s)-s^{n-1}f(0)-...-sf^{(n-2)}(0)-f^{(n-1)}(0)$
$s$-domain integral $\displaystyle\frac{f(t)}{t}$ $\displaystyle\int_s^{+\infty}F(\sigma)d\sigma$
$s$-domain first derivative $tf(t)$ $-F'(s)$
$s$-domain second derivative $t^2f(t)$ $F''(s)$
$s$-domain $n^{th}$ derivative $t^nf(t)$ $(-1)^nF^{(n)}(s)$

Common transform pairs ― The table below sums up the most common Laplace transform pairs:

$t$-domain $s$-domain
$a$ $\displaystyle\frac{a}{s}$
$t$ $\displaystyle\frac{1}{s^2}$
$t^n$ $\displaystyle\frac{n!}{s^{n+1}}$
$e^{at}$ $\displaystyle\frac{1}{s-a}$
$\cos(\omega t)$ $\displaystyle\frac{s}{s^2+\omega^2}$
$\sin(\omega t)$ $\displaystyle\frac{\omega}{s^2+\omega^2}$
$\textrm{cosh}(at)$ $\displaystyle\frac{s}{s^2-a^2}$
$\textrm{sinh}(at)$ $\displaystyle\frac{a}{s^2-a^2}$

Main operations ― The table below sums up the main operations of the Laplace transform:

Operation $t$-domain $s$-domain
Unit step function $u(t-a)$ $\frac{e^{-as}}{s}$
Dirac delta function $\delta(t-a)$ $e^{-as}$
$s$-shift $e^{at}f(t)$ $F(s-a)$
$t$-shift $u(t-a)f(t-a)$ $e^{-as}F(s)$