기록

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}(..

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=(..