# CS2001 工程數學 Engineering Mathematics
Textbook: Kreyszig, E. *Advanced Engineering Mathematics*. John Wiley & Sons, Inc.
Instructor: 賓拿雅 Binayak Kar
# Ordinary Differential Equations (ODEs)
* ODE: differential equations that depend on a single variable
* Modeling: a crucial feneral process that translates a physical situation or some other observations into a "mathematical model"
## 第１章 一階常微分方程式
### Sec. 1.1 Basic Concepts. Modeling
* differential equation: an equation containing derivatives of an unknown function
* order $n$: $n$th derivative of the unknown function y is the highest derivative
* first-order ODEs
* $F(x,y,y')=0$
* $y'=f(x,y)$
#### Concept of Solution
* solution: is defined and differentiable thoughout the interval and is such that the equation becomes an identity if $y$ and $y'$ are replaced by $h$ and $h'$
* solution curve: the curve of $h$
* ***the*** general solution: solution containing an arbitrary constant $c$
* develop methods that will gave general solution *uniquely*
* particular solution: choose a specific $c$
#### Initial Value Problem
* a problem with initial condition
* initial condition $y(x_0)=y_0$
#### More on Modeling
* STATE 1: Physical System
* Step 1: Modeling
* transit from the physical situation to mathematical method
* physical situation: engineering problem, physical problem
* mathmatical expressions
* variables
* functions
* equations
* STATE 2: Mathematical Model
* Step 2: Solving
* solve by mathematical method
* ODEs
* STATE 3: Mathematical Solution
* (Varify the solution)
* *Step 3: Interpreting*
* interpret result to physical
* STATE 4: Physical Interpretation
### Sec. 1.2 Geometric Meaning of $y'=f(x,y)$. Direction Fields, Euler's Method
- $y'(x_0)=f(x_0,y_0)$
- Graphic Method of Direction Fields
- show direction of solution curves of a given ODE by drawing short stright-line segments in the $xy$-plane
- isocline: curve of equal inclination
#### Numeric Method by Euler
- $y_n=y_{n-1}+hf(x_{n-1},y_{n-1})$
### Sec. 1.3 Separable ODEs. Modeling
- method of separating variables
1. $g(y)y'=f(x)$ (separable equation)
2. $\int g(y)y'dx=\int f(x)dx+c$
3. $\int g(y)dy=\int f(x)dx+c$
- $f$ and $g$ must be continuous
#### Extended Method: Reduction to Separable Form
- special case: $y'=f(\frac yx)$ (sometimes called homogeneous ODEs)
1. Let $y=ux$ and $y'=u'x+u$
2. $u'x+u=f(u)$
3. $\dfrac{du}{f(u)-u}=\dfrac{dx}x$
### Sec. 1.4 Exact ODEs. Integrating Factor
- **exact differential equation:** $M(x,y)dx+N(x,y)dy=0$ and differential form $du=\dfrac{\partial u}{\partial x}dx+\dfrac{\partial u}{\partial y}dy=0$
- **implicit solution** $u(x,y)=c$
1. check differential form: $\dfrac{\partial M}{\partial y}=\dfrac{\partial^2u}{\partial x\partial y}=\dfrac{\partial N}{\partial x}$
2. $u=\int Mdx+k(y)=\int Ndy+l(x)$
3. $\dfrac{\partial u}{\partial y}=\dfrac\partial{\partial y}\int Mdx+\dfrac{dk}{dy}=N$ or
$\dfrac{\partial u}{\partial x}=\dfrac\partial{\partial x}\int Ndy+\dfrac{dl}{dx}=M$
4. $k=\int dk=\int(N-\dfrac\partial{\partial y}\int Mdx)dy$ or
$l=\int dl=\int(M-\dfrac\partial{\partial x}\int Ndy)dx$
#### Reduction to Exact Form. Integrating Factors
- if differential form is not exact?
- multiply the nonexact equation
- $FPdx+FQdy=0$
- $F$ is called **integrating factor**
#### How to Find Integrating Factors
1. $R=\dfrac1Q(\dfrac{\partial P}{\partial y}-\dfrac{\partial Q}{\partial x})$
2. if $R$ depends only on $x$, $F=e^{\int Rdx}$
if $R$ depends only on $y$, $F=e^{\int Rdy}$
### Sec. 1.5 Linear ODEs. Bernoulli Equation. Population Dynamics
- **linear:** if can brought into the $y'+p(x)y=r(x)$ form
- if not, it is nonlinear
- $p$ and $r$ may be ***any*** functions of $x$
- In enginerring, $r$ is called the input, $y(x)$ is called output or response
- **Homogeneous Linear ODE:** $y'+p(x)y=0$
- general solution: $y(x)=ce^{-\int p(x)dx}$
- trivial solution: $y(x)=0$
- **Nonhomogeneous Linear ODE:**
- multiply the nonhomogeneous equation
- $F=\int pdx$
- general solution: $y(x)=F^{-1}(\int Frdx+c)$
- Total Output = Response to the Input + Response to the Initial Data
#### Reduction to Linear Form. Bernoulli Equation
- **Bernoulli equation:** $y'+p(x)y=g(x)y^a$
- if $a=0$ or $a=1$, equation is linear
- set $u(x)=[y(x)]^{1-a}$
- get $u'+(1-a)pu=(1-a)g$
#### Population Dynamics
## 第２章 二階線性常微分方程式
### Sec. 2.1 Homogeneous Linear ODEs of Second Order
* **linear:** $y''+p(x)y'+q(x)y=r(x)$
* **homogenous:** $r(x)=0$
* $p$, $q$: coefficients
* soulution: $y=h(x)$
#### Homogeneous Linear ODEs: Superposition Principle
* **superposition principle**/**linearity principle**
* for a homogeneous linear ODE, any linear combination of two solutions on an open interval I is again a solution on interval I.
* sums and constant multiples of solutions also are solutions.
* PROOF
* let $y_1$ and $y_2$ be solution on I.
* substitute $y=c_1y_1+c_2y_2$
* we get $y''+py'+qy=(c_1y_1+c_2y_2)''+p(c_1y_1+c_2y_2)'+q(c_1y_1+c_2y_2)$
$=c_1y_1''+c_2y_2''+p(c_1y_1'+c_2y_2')+q(c_1y_1+c_2y_2)$
$=c_1(y_1''+py_1'+qy_1)+c_2(y_2''+py_2'+qy_2)=0$
* this theorem **does not hold** for *++nonhomogeneous++ linear* or *++nonlinear++* ODEs
#### Initial Value Problem. Basis. General Solution
* two initial conditions: $y(x_0)=K_0$ , $y'(x_0)=K_1$
* use condition to determine $c_1$ and $c_2$, get a **particular solution**
### Sec. 2.2 Homogeneous Lineat ODEs with Constant Coefficients
* $y''+ay'+by=0$
* $a$ and $b$ are constant
*
### Sec. 2.3 Differential Operators. *Optional*
補
### Sec. 2.5 Euler-Cauchy Equations
1. **Euler-Cauchy equation** (problem) $x^2y''+axy'+by=0$
2. substitute $y=x^m$
3. **auxiliary equation** $m^2+(a-1)m+b=0$
* **Case Ⅰ.** Real different roots $m=m_1,m_2$
* $y=c_1x^{m_1}+c_2x^{m_2}$
* **Case Ⅱ.** A real double root $m=m$
* $y=(c_1+c_2\ln x)x^m$
* Case Ⅲ. Complex conjugate roots $m=r\pm\omega i$
* $y=x^r[A\cos(\omega\ln x)+B\sin(\omega\ln x)]$
### Sec. 2.6 Existence and Uniqueness of Solutions. Wronskian
#### Linear Independence of Solutions
* linear independent: $k_1y_1(x)+k_2y_2(x)=0$
* linear dependent: $y_1=ky_2$
* **Linear Dependence and Independence of Solutions**
* Let the homogeneous linear ODE have continuous coefficients $p(x)$ and $q(x)$ on an open interval $I$. Then two solutions $y_1$ and $y_2$ of the ODE on $I$ are linearly dependent on $I$ iif their *Wronski determinant* **(Wronskian)** $W(y_1,y_2)=\left|\begin{matrix}y_1&y_2\\y_1'&y_2'\\\end{matrix}\right|=y_1y_2'-y_2y_1'=0$ ; hence, *if there is an $x1$ in $I$ at which $W$ is not 0, then $y1,y2$ are linearly independent on $I$*
#### A General Solution of (1) Includes All Solutions
### Sec. 2.7 Nonhomogeneous ODEs
* $y''+p(x)y'+q(x)y=r(x)$
* general solution: $y(x)=y_h(x)+y_p(x)$
* $y_h(x)=c_1y_1(x)+c_2y_2(x)$ (general solution of the homogeneous ODE $r(x)=0$)
* $y_p(x)$ particular solution
#### Method of Undertermined Coefficients
| Term in $r(x)$ | Choice for $y_p(x)$ |
| -------------------------------------------------------- | ------------------------------------------- |
| $ke^{\gamma x}$ | $Ce^{\gamma x}$ |
| $kx^n$ | $K_nx^n+K_{n-1}x^{n-1}+...+K_1x+K_0$ |
| $k\cos\omega x$, $k\sin\omega x$ | $K\cos\omega x+M\sin\omega x$ |
| $ke^{\alpha x}\cos\omega x$, $ke^{\alpha x}\sin\omega x$ | $e^{\alpha x}(K\cos\omega x+M\sin\omega x)$ |
* Choice Rules
* **Basic Rule**
* **Modification Rule** (as double root)
* **Sum Rule** (sum of two choice)
### Sec. 2.10 Solution by Variation of Parameters
* $y_p(x)=-y_1\int\dfrac{y_2r}Wdx+y_2\int\dfrac{y_1r}Wdx$
#### Idea of the Method. Derivation of (2)
證明，補
## 第３章 高階線性常微分方程式
### Sec. 3.1 Homogeneous Linear ODEs
### Sec. 3.2 Homogeneous Linear ODEs with Constant Coefficients
### Sec. 3.3 Nonhomogeneous Linear ODEs
## 第６章 拉普拉斯轉換