Second-Order Linear ODE (2) - Constant Coefficients

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=(\lambda -\lambda_{1} )(\lambda -\lambda_{2} )=0 $

$ e^{\lambda_{1}x}, e^{\lambda_{2}x} $: Linearly Independent

$ \therefore y=c_{1}e^{\lambda_{1}x}+c_{2}e^{\lambda_{2}x} $

 

2) Real Double Root

$ \lambda^{2}+a\lambda+b=\lambda^{2}+a\lambda+\frac{a^{2}}{4}=(\lambda +\frac{1}{2}a)^{2}=0 $

$ \lambda =-\frac{1}{2}a $

$ y_{1}=e^{-\frac{1}{2}ax} $

$ y_{2}=uy_{1}, u=\int U dx $

$ U=\frac{1}{y_{1}^{2}}e^{-\int a dx}=e^{ax}e^{-ax}=1 $

$ u=\int U dx=\int 1 dx=x $

$ y_{2}=uy_{1}=xe^{-\frac{1}{2}ax} $

$ \therefore y=c_{1}y_{1}+c_{2}y_{2}=c_{1}e^{-\frac{1}{2}ax}+c_{2}xe^{-\frac{1}{2}ax}=(c_{1}+c_{2}x)e^{-\frac{1}{2}ax} $

 

ex) $ y''+3y'+2.25y=0 $

$ y=e^{\lambda x} \rightarrow e^{\lambda x}(\lambda^{2}+3\lambda+2.25)=0 $

$ (\lambda +1.5^{2})=0 $

$ \lambda =-1.5 $

$ \therefore y=(c_{1}+c_{2}x)e^{-1.5x} $

 

3) Complex Root

$ \lambda^{2}+a\lambda+b=(\lambda -(m+ni))(\lambda -(m-ni))=0 $

$ e^{(m+ni)x}, e^{(m-ni)x} $: Linearly Independent

$ y=c_{1}e^{(m+ni)x}+c_{2}e^{(m-ni)x} $

   $ =c_{1}e^{mx}e^{nix}+c_{2}e^{mx}e^{-nix} $

   $ =c_{1}e^{mx}(\cos nx+i\sin nx)+c_{2}e^{mx}(\cos nx-i\sin nx) $ (by Euler Formula)

   $ =e^{mx}((c_{1}+c_{2})\cos nx+(c_{1}i-c_{2}i)\sin nx) $

   $ =e^{mx}(A\cos nx+B\sin nx) $

 

 

5. Euler Formula

 

$ e^{it}=\cos t+i\sin t $

$ e^{x}=1+x+\frac{x^{2}}{2!}+... $

$ e^{it}=1+it+\frac{(it)^{2}}{2!}+...=(1-\frac{t^{2}}{2!}+\frac{t^{4}}{4!}-\frac{t^{6}}{6!}+...)+i(t-\frac{t^{3}}{3!}+\frac{t^{5}}{5!}-...)=\cos t+i\sin t $