$\newcommand{\dis}{\displaystyle} \newcommand{\m}{\hspace{1em}} \newcommand{\mm}{\hspace{2em}} \newcommand{\x}{\vspace*{1ex}} \newcommand{\xx}{\vspace*{2ex}} \let\limm\lim \renewcommand{\lim}{\dis\limm} \let\fracc\frac \renewcommand{\frac}{\dis\fracc} \let\summ\sum \renewcommand{\sum}{\dis\summ} \let\intt\int \renewcommand{\int}{\dis\intt} \newcommand{\sech}{\text{sech}} \newcommand{\csch}{\text{csch}} \newcommand{\Ln}{\text{Ln}} \newcommand{\p}{\partial} $

Lecture 18, 10/3/2022. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A

  1. Change of variables to solve equations
    1. Use a combination of differentials and chain rules;
    2. Reduce to an equation that can be solved.
  2. Leibniz' rule
    1. $\int_a^x f(t)dt = F(t)\Big|_a^x = F(x) - F(a)$
    2. $ \frac{d}{dx}\int_a^x f(t)dt = f(x)$
    3. $ \frac{d}{dx}\int_x^b f(t)dt = -f(x)$
    4. $ \frac{d}{dx}\int_{u(x)}^{v(x)} f(t)dt = f(v(x))\frac{dv}{dx} -f( u(x)) \frac{du}{dx} $
    5. $ \frac{d}{dx}\int_{u(x)}^{v(x)} f(x,t)dt = f(x,v(x))\frac{dv}{dx} -f(x, u(x)) \frac{du}{dx} + \int_{u(x)}^{v(x)} \frac{\p f}{\p x}(x,t)dt $