기록

9. Existence and Uniqueness of Solution $ y''+py'+qy=0 $ $ y(x_{0})=K_{0}, y'(x_{0})=K_{1} $ Theorem 1) Existence and Uniqueness Theorem for Initial Value Problems p, q: continuous on I ($ x_{0}\in I $) Then $ y''+py'+qy=0 $ ($ y(x_{0})=K_{0}, y'(x_{0})=K_{1} $) has a unique solution on I. Theorem 2) Linear Dependence and Independence of Solutions i) $ y_{1}, y_{2} $: Linearly Dependent $ \Leftr..

8. Euler-Cauchy Equation $ x^{2}y''+axy'+by=0 $ $ y=x^{m} \rightarrow y'=mx^{m-1}, y''=m(m-1)x^{m-2} $ $ \rightarrow x^{2}m(m-1)x^{m-2}+axmx^{m-1}+bx^{m}=0 $ $ x^{m}(m^{2}+(a-1)m+b)=0 $ $ m^{2}+(a-1)m+b=0 $ 1) Real Different Roots $ m=m_{1}, m_{2} $ $ x^{m_{1}}, x^{m_{2}} $: Linearly Independent $ \therefore y=c_{1}x^{m_{1}}+c_{2}x^{m_{2}} $ ex) $ x^{2}y''+xy'-9y=0 $ $ y=x^{m} \rightarrow x^{m}(..

6. Free Oscillation of a Mass Spring System - Undamped Small Damping & Short Time → Disregard Damping $ my''+ky=0 $ (k: spring constant) $ y''+\frac{k}{m}y=0 $ $ y''+\omega _{0}^{2}y=0 $ ($ \omega _{0}=\sqrt{\frac{k}{m}} $) $ y=e^{\lambda t} \rightarrow \lambda ^{2}+\omega _{0}^{2}=0 \rightarrow \lambda =\pm \omega _{0}i $ $ \therefore y(t)=A\cos \omega _{0}t+B\sin \omega _{0}t $ $ =\sqrt{A^{2}+..

4. Second-Order Linear ODE with Constant Coefficients $ y''+ay'+by=0 $ $ y=e^{\lambda x} \rightarrow y'=\lambda e^{\lambda x}, y''=\lambda^{2} e^{\lambda x} $ $ \lambda^{2} e^{\lambda x}+a\lambda e^{\lambda x}+be^{\lambda x} $ $ =(\lambda^{2}+a\lambda+b)e^{\lambda x}=0 $ $ \lambda^{2}+a\lambda+b=0 $: Characteristic Equation(Auxiliary Equation) 1) Two Distinct Real Root $ \lambda^{2}+a\lambda+b=(..

1. Homogeneous Linear ODE of Second Order $ y''+p(x)y'+q(x)y=r(x)\left\{\begin{matrix} r=0 (homo) \\ r\neq 0 (non-homo) \end{matrix}\right. $ $ L=\frac{\mathrm{d}^{2} }{\mathrm{d} x^{2}}+p\frac{\mathrm{d} }{\mathrm{d} x}+q $ $ L(y)=r(x) $ $ L(\alpha y_{1}+\beta y_{2})=(\alpha y_{1}+\beta y_{2})''+p(\alpha y_{1}+\beta y_{2})'+q(\alpha y_{1}+\beta y_{2}) $ $ =\alpha y_{1}''+\beta y_{2}''+p\alpha y..

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..

50. Vector Integration $ \vec{A}=\vec{A}(x_{i}) $ 1) Volume Integral $ \int_{V}^{}\vec{A}dv=(\int_{V}^{}A_{1}dv, \int_{V}^{}A_{2}dv, \int_{V}^{}A_{3}dv) $ 2) Surface Integral $ \int_{S}^{}\vec{A}\cdot d\vec{a}=\int_{S}^{}\vec{A}\cdot \hat{n}da=\int_{S}^{}\sum_{i}^{}A_{i}da_{i} $ (Outward Normal is positive.) 3) Line Integral $ \int_{BC}^{}\vec{A}\cdot d\vec{s}=\int_{BC}^{}\sum_{i}^{}A_{i}dx_{i} ..

35. Gradient Operator (Vector Differential Operator) Scalar Function $ \phi (x_{1},x_{2},x_{3}) $ $ \phi' (x_{1}',x_{2}',x_{3}')=\phi (x_{1},x_{2},x_{3}) $: Coordinate Transformation $ \frac{\partial \phi '}{\partial x_{i}'}=\sum_{j}^{}\frac{\partial \phi }{\partial x_{j}}\frac{\partial x_{j}}{\partial x_{i}'} $: Chain Rule $ x_{j}=\sum_{k}^{}\lambda _{kj}x_{k}' $: Inverse Coordinate Transformat..