Question

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

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

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