Suppose I have 3 equations:\begin{eqnarray} y=Ae^{m_1x}+Be^{m_2x}\\ y^\prime=Am_1e^{m_1x}+Bm_2e^{m_2x}\\ y^{\prime\prime}=A{m_1}^2e^{m_1x}+B{m_2}^2e^{m_2x} \end{eqnarray} Here $m_1$ and $m_2$ are fixed constants. I need to eliminate $A$ and $B$ from the above three equations. One thing I can do is to solve for $A$ and $B$ from the last two equations and replace their values in the first equation, but that's a very tedious process. I am looking for a simpler method. Any help appreciated. Thanks!
P.S: One thing I noticed is that the answer looks like this: $y^{\prime\prime}-(m_1+m_2)y^\prime+(m_1m_2)y=0$. Notice that this is somewhat similar to the polynomial equation: $x^2-(m_1+m_2)x+(m_1m_2)=0$
Where $m_1$ and $m_2$ are solutions for $x$.