기록

6. Linear $ f(\alpha x+\beta y)=\alpha f(x)+\beta f(y) $: Linear $ y'+p(x)y=r(x) \left\{\begin{matrix} r=0 (homo)\\ r\neq 0 (non-homo) \end{matrix}\right. $ $ L=\frac{\mathrm{d} }{\mathrm{d} x}+p $ $ L(y)=r(x) $ $ L(\alpha y_{1}+\beta y_{2})=\frac{\mathrm{d} }{\mathrm{d} x}(\alpha y_{1}+\beta y_{2})+p(\alpha y_{1}+\beta y_{2})=\alpha y_{1}'+\beta y_{2}'+p\alpha y_{1}+p\beta y_{2} $ $ =\alpha (y_..

4. Exact ODE Total Differential $ u(x,y)=c $ $ du=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy=u_{x}dx+u_{y}dy=0 $ Definition $ M(x,y)dx+N(x,y)dy=0 $ is called an Exact ODE $ \exists u(x,y) $ s.t. $ u_{x}=M, u_{y}=N $ Theorem $ M(x,y)dx+N(x,y)dy=0 $ is exact $ \Leftrightarrow M_{y}=N_{x} $ $ u=\int M dx+k(y)=\int N dy+l(x) $ ex) $ (e^{x}+y)dx+(x-e^{-y})dy=0 $ $ M=e^{x}+y, N=x-..

Ordinary Differential Equation 카테고리는 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..