Find the general solution of CauchyEuler Equation x2y 5xy

Find the general solution of Cauchy-Euler Equation x^2y\" - 5xy + 9y = 0, x > 0.

Solution

We assume that y=x^m.

Differentiating,we have dy/dx=mx^m-1 and d²y/dx²=m(m-1)x^m-2

^{Substituting into the original equation ,we have}

x² (m(m-1)x^m-2)-5xm(x^m-1)+9x^m=0

^{On rearranging gives}

(m²x²-mx²)x^m-2-5xmx^m-1+9x^m=0

(m(m-1)-5m+9)x^m=0

m²-m-5m+9=0

m²-6m+9=0

m²-3m-3m+9=0

(m-3)(m-3)=0

m=3,3

Therefore roots are 3,3

Therefore general solution is y=c₁x³+c₂x³.