기록

6. with Auxiliary Conditions $ g=\sum_{i}x_{i}^{2}-\rho ^{2}=0 \rightarrow r=\rho =constant $: Equations of Constraint (구면 위를 따라간다) $ f=f(y_{i},y_{i}';x)=f(y,y',z,z';x) $ $ g(y_{i};x)=g(y,z;x)=0 $ $ \Rightarrow \frac{\partial y}{\partial \alpha }, \frac{\partial z}{\partial \alpha } $ are not independent (하나의 variable이 바뀌면, 다른 variable도 바뀐다) $ \therefore $ for $ \alpha =0, \frac{\partial y}{\par..

4. 2nd Form of the Euler Equation to get better form of equation for $ \frac{\partial f}{\partial x}=0 $ $ df=\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial y'}dy'+\frac{\partial f}{\partial x}dx $ $ \frac{\mathrm{d} f}{\mathrm{d} x}=\frac{\mathrm{d} }{\mathrm{d} x}f(y,y';x)=\frac{\partial f}{\partial y}\frac{\mathrm{d} y}{\mathrm{d} x}+\frac{\partial f}{\partial y'}\frac{\mathrm{d} ..

2. Euler's Equation $ J=\int_{x_{1}}^{x_{2}}f(y(\alpha ,x),y'(\alpha ,x);x)dx $ $ \frac{\partial J}{\partial \alpha }=\frac{\partial }{\partial \alpha }\int_{x_{1}}^{x_{2}}f(y,y';x)dx $ $ =\int_{x_{1}}^{x_{2}}(\frac{\partial f}{\partial y}\frac{\partial y}{\partial \alpha }+\frac{\partial f}{\partial y'}\frac{\partial y'}{\partial \alpha })dx $ ($ \because y(\alpha ,x)=y(0,x)+\alpha \eta (x) \ri..

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. 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_..