Step 1

Solving the equation

\(\displaystyle{\left({4}{D}^{{{4}}}-{4}{D}^{{{3}}}-{23}{D}^{{{2}}}+{12}{D}+{36}\right)}{y}={0}\)

With the aid synthetic division, it is easily seen that the auxiliary equation

\(\displaystyle{4}{m}^{{{4}}}-{4}{m}^{{{3}}}-{23}{m}^{{{2}}}+{12}{m}+{36}={0}\)

\(\displaystyle{\left({m}-{2}\right)}^{{{2}}}{\left({2}{m}+{3}\right)}^{{{2}}}={0}\ \ \ {\left({F}{a}{c}\to{r}\right)}\)

\(m-2=0\ or\ m-2=0\ or\ 2m+3=0\ or\ 2m+3=0\)

(Set each factor equal to 0)

\(m=2\) or \(m=2\) or \(\displaystyle{m}=-{\frac{{{3}}}{{{2}}}}{\quad\text{or}\quad}{m}=-{\frac{{{3}}}{{{2}}}}\)

Step 2

Whose roots \(m=2,2\),\(\displaystyle-{\frac{{{3}}}{{{2}}}},-{\frac{{{3}}}{{{2}}}}\) may be obtained by synthetic division. Then the general solution is seen to be

\(\displaystyle{y}={\left({c}_{{{1}}}+{c}_{{{2}}}{x}\right)}{e}^{{{2}{x}}}+{\left({c}_{{{3}}}+{c}_{{{4}}}{x}\right)}{e}^{{{\frac{{-{3}{x}}}{{{2}}}}}}\)

Therefore the solution is \(\displaystyle{y}={\left({c}_{{{1}}}+{c}_{{{2}}}{x}\right)}{e}^{{{2}{x}}}+{\left({c}_{{{3}}}+{c}_{{{4}}}{x}\right)}{e}^{{{\frac{{-{3}{x}}}{{{2}}}}}}\)