CONTENTS

• Basic Concepts, Modeling
• Direction Fields, Euler's Method ( Geometric Meaning of $y^,=f(x,y)$ )
• Separable ODEs, Reduction to Separable Form
• Exact ODEs, Integrating Fcators
• Linear ODEs, Bernoulli Equation
• Othogonal Trajectories
• Existence and Uniqueness of Soultions for I.V.P.

1.1 Basic Concepts, Modeling

Mathmatical Modeling

A model is very often an equation containing derivatives of an unknown function.
Such a model is called a Differential Equation.

D.E ( Differential Equation )

1. ODE ( Ordinary Differential Equation )
   An equation that contains one or several derivatives of an unknown function of one independent variable.
   such as $y'=cosx$ , $y''+9y=0$ , $x^2y'''y'+2e^xy''=(x^2+2)y^2$
2. PDE ( Partial Differential Equation )
   An equation involving partial derivatives of an unknown function of two or more variables.
   such as $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$

Solution

$y=h(x)$ is called a solution if it satisfies the ODE in a given interval $a<x<b$ (open interval /closed interval). → explicit solution

$u(x,y)=0$ → implicit solution

Solution
1. General Solution : a solution containing an arbitrary constant ( $C$ ).
2. Particular Solution : a solution that we choose a specific constant ( $C=3$ )
3. Singular Solution : an additional solution that cannot be obtained from the general solution.

such as ODE $(y')^2-xy+y=0$ ,
G.S : $y=cx-c^2$
P.S : $y=2x-4$
S.S : $y=x^2/4$

Initial Value Problem (I.V.P)

General Soultion $+$ Initial Conditions(I.Cs) $=$ Particular Solution

1.3 Separable ODEs, Reduction to Separable Form

$g(y)*y'=f(x)$
: Separable Equation

Method of Separating Variables

$g(y)*y'=f(x)$
→ $g(y)\frac{dy}{dx}=f(x)$
→ $\int g(y) dy= \int f(x) dx$
→ $G(y)=F(x)+C$

Reduction to Separable Form

A homogeneous ODE $y'=f(\frac{y}{x})$ can be reduced to separabel form by the substitution of $y=ux$.

$y'=f(\frac{y}{x})$
→ $u'x+u=f(u)$
→ $\frac{du}{f(u)-u}=\frac{dx}{x}$
where $y=ux⇒ u=\frac{y}{x}$ , $y'=\frac{d(ux)'}{dx}=u'x+u$

$EX)$ $2xyy'=y^2-x^2$
$∴x^2+y^2=c^*x$

1.4 Exact ODEs, Integrating Factors

Exact Differential Equation

A first order ODE $M(x,y)+N(x,y)y'=0$ , written as $M(x,y)dx+N(x,y)dy=0$ is called an Exact Differential Equation if the differential form $M(x,y)dx+N(x,y)dy$ is exact, that is, this form is the differential $du=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy$ of some function $u(x,y)$. Then,
$M(x,y)dx+N(x,y)dy=0$
→ $du=0$
→ $u(x,y)=C$ : implicit solution

Condition for Exactness

Necessary and sufficient condition to be an exact differential equation.

$M(x,y)dx+N(x,y)dy=0$
→ $du=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy=0$
→ $\frac{du}{dx}=M, \frac{du}{dy}=N$
→ $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$ : Condition for Exactness

How to Solve

• Case 1
  $M(x,y)=\frac{\partial u}{\partial x}$
  ⇒ $u(x,y)=\int M(x,y)dx+k(y)$
  ⇒ $\frac{\partial u}{\partial y}=N(x,y)$
  ⇒ $\frac{dk}{dy}$
  ⇒ $k(y)$

• Case 2
  $N(x,y)=\frac{\partial u}{\partial y}$
  ⇒ $u(x,y)=\int N(x,y)dx+l(x)$
  ⇒ $\frac{\partial u}{\partial x}=M(x,y)$
  ⇒ $\frac{dl}{dx}$
  ⇒ $l(x)$

Integrating Factor

$(FP)dx+(FQ)dy=0$ , where $FP=M, FQ=N$
Condition for Exactness : $\frac{\partial (FP)}{\partial y}=\frac{\partial (FQ)}{\partial x}$

The function $F(x,y)$ is the solution that depends on both x and y.

1. $F=F(x)$ : function of x only
   This function depends soley on x , $\frac{\partial F}{\partial x}=F′, \frac{\partial F}{\partial y}=0$
   so, Conditon for Exactness be made for like
   $\frac{\partial (FP)}{\partial y}=\frac{\partial (FQ)}{\partial x}$

2. $F=F(y)$ : function of y only

link death

hello world! *hi*