First-Order ODE (1) - Separable ODE

Ordinary Differential Equation 카테고리는 <Advanced Engineering Mathematics>10th (Erwin Kreyszig)의 순서를 따라갑니다

 

 

1. Basic Concepts

 

Physical System → Mathematical Model → Mathematical Solution → Physical Interpretation

 

Ordinary Differential Equation
Order(n)
First-Order ODE: $ F(x,y,y')=0, y'=f(x,y) $

 

General Solution: containing an arbitrary constant c
Particular Solution: not containing an arbitrary constant c

 

Initial Value Problem: $ y'=f(x,y), y(x_{0})=y_{0} $

 

 

2. Separable ODE

 

$ y'=\frac{\mathrm{d} y}{\mathrm{d} x} $

 

$ g(y)y'=f(x) $

$ g(y)\frac{\mathrm{d} y}{\mathrm{d} x}=f(x) $

$ g(y){\mathrm{d} y}=f(x){\mathrm{d} x} $

$ \int g(y){\mathrm{d} y}=\int f(x){\mathrm{d} x}+c $

 

ex) $ y'=1+y^{2} $

$ \frac{{\mathrm{d} y}}{1+y^{2}}={\mathrm{d} x} $

$ \int \frac{{\mathrm{d} y}}{1+y^{2}}=\int {\mathrm{d} x} $

$ \tan^{-1}y=x+c $

$ \therefore y=\tan (x+c) $

 

 

3. Reduction to Separable Form

 

$ y'=f(\frac{y}{x}) $

 

$ \frac{y}{x}=u \rightarrow y=ux $

$ y'=u'x+u=f(u) $

$ u'x=f(u)-u $

$ \frac{{\mathrm{d} u}}{f(u)-u}=\frac{{\mathrm{d} x}}{x} $

$ \int \frac{{\mathrm{d} u}}{f(u)-u}=\int \frac{{\mathrm{d} x}}{x}+c $