Question

Solve the for the general solution of the differential equation (4D^{4}-4D^{3}-23D^{2}+12D+36)y=0

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ANSWERED
asked 2021-04-25
Solve the for the general solution of the differential equation
\(\displaystyle{\left({4}{D}^{{{4}}}-{4}{D}^{{{3}}}-{23}{D}^{{{2}}}+{12}{D}+{36}\right)}{y}={0}\)

Answers (1)

2021-04-27

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

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