기록

$ \int e^{ax}\sin (bx)dx=\frac{e^{a}(a\sin (bx)-b\cos (bx))}{b^{2}+a^{2}}+C $ $ \int e^{ax}\cos (bx)dx=\frac{e^{a}(b\sin (bx)+a\cos (bx))}{b^{2}+a^{2}}+C $ $ \int x\sin (ax)=\frac{\sin (ax)-ax\cos (ax)}{a^{2}}+C $ $ \int x\cos (ax)=\frac{ax\sin (ax)+\cos (ax)}{a^{2}}+C $

7. Delta Notation $ \frac{\partial J}{\partial \alpha }d\alpha =\int_{x_{1}}^{x_{2}}(\frac{\partial f}{\partial y}-\frac{\mathrm{d} }{\mathrm{d} x}\frac{\partial f}{\partial y'})\frac{\partial y}{\partial \alpha }d\alpha dx $ $ \Rightarrow \delta J=\int_{x_{1}}^{x_{2}}(\frac{\partial f}{\partial y}-\frac{\mathrm{d} }{\mathrm{d} x}\frac{\partial f}{\partial y'})\delta y dx $ ($ \frac{\partial J}{..

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..

5. Functions with Several Dependent Variables $ f=f(y_{1}(x),y_{1}'(x),y_{2}(x),y_{2}'(x), ... ;x) $ $ \Rightarrow f=f(y_{i}(x),y_{i}'(x);x), i=1,2,...,m $ $ y_{i}(\alpha ,x)=y_{i}(0,x)+\alpha \eta _{i}(x) $ $ \frac{\partial J}{\partial \alpha }=\int_{x_{1}}^{x_{2}}\sum_{i}(\frac{\partial f}{\partial y_{i}}-\frac{\mathrm{d} }{\mathrm{d} x}\frac{\partial f}{\partial y_{i}'})\eta _{i}(x) dx=0 $ $ ..

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..

Calculus of Variation 카테고리는 5th (Marion)의 chapter 6 순서를 따라갑니다 1. Calculus of Variation find an extreme solution (max or min) $ J=\int_{x_{1}}^{x_{2}}f(y(x),y'(x);x)dx $ ($ y'\equiv \frac{\mathrm{d} y}{\mathrm{d} x} $ ) (x: independent variable, y: dependent variable) $ y=y(\alpha ,x) $ $ \alpha =0 \rightarrow y=y(0,x)=y(x) $: function that yields an extreme for J $ \Rightarrow y=y(\alpha ,x)=y(0..

15. Variation of Parameters $ y''+py'+qy=r $ $ y(x)=y_{h}(x)+y_{p}(x)=(c_{1}y_{1}(x)+c_{2}y_{2}(x))+y_{p}(x) $ $ y_{1}''+py_{1}'+qy_{1}=0, y_{2}''+py_{2}'+qy_{2}=0 $ $ y_{p}=uy_{1}+vy_{2} $ $ y_{p}'=u'y_{1}+uy_{1}'+v'y_{2}+vy_{2}'=uy_{1}'+vy_{2}'(u'y_{1}+v'y_{2}=0) $ $ y_{p}''=u'y_{1}'+uy_{1}''+v'y_{2}'+vy_{2}'' $ $ (u'y_{1}'+uy_{1}''+v'y_{2}'+vy_{2}'')+p(uy_{1}'+vy_{2}')+q(uy_{1}+vy_{2}) $ $ =u..

12. Driving Force $ my''+cy'+ky=r(t)=F_{0}cos\omega t $ $ r(t)=F_{0}cos\omega t $: Driving Force $ y=y_{h}+y_{p} $ $ y_{p}=a\cos \omega t+b\sin \omega t $ $ y_{p}'=-\omega a\sin \omega t+\omega b\cos \omega t $ $ y_{p}''=-\omega ^{2}a\cos \omega t-\omega ^{2}b\sin \omega t $ $ \rightarrow m(-\omega ^{2}a\cos \omega t-\omega ^{2}b\sin \omega t)+c(-\omega a\sin \omega t+\omega b\cos \omega t)+k(a\..

10. Definition and Theorems $ y''+p(x)y'+q(x)y=r(x) $ Definition) A general Solution of $ y''+p(x)y'+q(x)y=r(x) $ $ \Rightarrow y=y_{h}+y_{p}=(c_{1}y_{1}+c_{2}y_{2})+y_{p} $ ($ y_{h} $: solution of $ y''+py'+qy=0 $) Theorem 1) $ y''+py'+qy=0 $: a $ y''+py'+qy=r $: b i) a의 해 + b의 해 = b의 해 ii) b의 해 - b의 해 = a의 해 $ Y_{1} $: a의 해 $ \rightarrow Y_{1}''+pY_{1}'+qY_{1}=0 $ $ Y_{2} $: b의 해 $ \rightarrow..