{F} Find the solution of
(D^3+3D^2-4)y=0
The auxiliary equation is m3+3m2−4=0m^3+3m^2-4=0m3+3m2−4=0
Performing synthetic division,
1130−4144144∣0\begin{array}{c|rrr} 1&1&3&0&-4\\ & & 1 & 4&4\\ \hline &1&4&4&\mid 0 \end{array}111314044−44∣0
We get m−1m-1m−1 as a factor. Therefore,
m3+3m2−4=(m−1)(m2+4m+4)=0 ⟹ m=1 & m2+4m+4=0i.e.,m=1 & m=−2(twice)m^3+3m^2-4= (m-1)(m^2+4m+4)=0\\ \implies m=1~\&~ m^2+4m+4=0\\ i.e., m=1~\&~ m=-2(twice)\\m3+3m2−4=(m−1)(m2+4m+4)=0⟹m=1 & m2+4m+4=0i.e.,m=1 & m=−2(twice)
The solution is y=c1ex+(c2+c3x)e−2xy = c_1 e^x + (c_2+c_3x)e^{-2x}y=c1ex+(c2+c3x)e−2x
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