Second-Order Linear ODE (4) - Euler-Cauchy Equation

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}(m(m-1)+m-9)=0 $

$ m^{2}=9 $

$ m=\pm 3 $

$ \therefore y=c_{1}x^{3}+c_{2}x^{-3} $

 

2) Real Double Roots

$ m^{2}+(a-1)m+b=0 $

$ b=\frac{(a-1)^{2}}{4} $

$ \rightarrow (m+\frac{a-1}{2})^{2}=0 $

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

$ \Rightarrow y_{1}=x^{\frac{1-a}{2}} $

$ y_{2}=uy_{1}, u=\int U dx $ ($ y''+\frac{a}{x}y'+\frac{b}{x^{2}}y=0 $)

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

$ u=\int \frac{1}{x} dx=\ln x $

$ \Rightarrow y_{2}=uy_{1}=\ln x \cdot x^{\frac{1-a}{2}} $

$ \therefore y=c_{1}x^{\frac{1-a}{2}}+c_{2}\ln x \cdot x^{\frac{1-a}{2}}=(c_{1}+c_{2}\ln x)x^{\frac{1-a}{2}} $

 

3) Complex Conjugate Roots

$ m=\alpha \pm \beta i $

$ \therefore y=c_{1}x^{\alpha +\beta i}+c_{2}x^{\alpha -\beta i} $

   $ =c_{1}x^{\alpha}x^{\beta i}+c_{2}x^{\alpha}x^{-\beta i} $

   $ =c_{1}x^{\alpha}e^{\ln x^{\beta i}}+c_{2}x^{\alpha}e^{\ln x^{-\beta i}} $

   $ =c_{1}x^{\alpha}e^{\beta i\ln x}+c_{2}x^{\alpha}e^{-\beta i\ln x} $

   $ =c_{1}x^{\alpha}e^{i(\beta \ln x)}+c_{2}x^{\alpha}e^{i(-\beta \ln x)} $

   $ =c_{1}x^{\alpha}(\cos (\beta \ln x)+i\sin (\beta \ln x))+c_{2}x^{\alpha}(\cos (\beta \ln x)-i\sin (\beta \ln x)) $

   $ =x^{\alpha}((c_{1}+c_{2})\cos (\beta \ln x)+(c_{1}i-c_{2}i)\sin (\beta \ln x)) $

   $ =x^{\alpha}(A\cos (\beta \ln x)+B\sin (\beta \ln x)) $