기록

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..

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_..