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The "Euler Equation" Method for non-constant-coefficients 2nd order linear homogeneous ODE

If the ODE is like or can be transformed to \(x^{2}y''+bxy'+cy=0\) then it can use the "Euler Equation" method:
1. Try ansatz: \(y=x^{p}\), \(p\) is an unknown constant.
2. Plug it in ODE to get the auxiliary equation \(p(p-1)+ap+b=0\).
3. Using the solution of the auxiliary equation \(p_{1},p_{2}\) to solve the ODE: \(y_{general}(x)=Ax^{p_{1}}+Bx^{p_{2}}\) , where \(A,B=\) constant.
- If \(p_{1}=p_{2}=p\), claim: \(y_{general}(x)=(A+B\ln x)x^{p}\)

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