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The general form of 2nd-order linear homogeneous ODE is \(a_{2}(x)\frac{d^{2}y}{dx^{2}}+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y(x)=0\)

If \(a_{2}(x),a_{1}(x),a_{0}(x)\) are constant functions:

1. Try ansatz: \(y=e^{mx}\).

2. To find \(m\), plug \(y = e^{mx}\) into ODE. Rearrange the equation and get the auxiliary equation: \(\underbrace{(am^{2}+bm+c)}_{\text{Auxiliary Equation}}e^{mx}=0\)

3. Using the solution of the auxiliary equation to solve the ODE.

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