solve the Differential equations y'''+10y''+25y'=0

Question
Differential equations
solve the Differential equations $$\displaystyle{y}{'''}+{10}{y}{''}+{25}{y}'={0}$$

2020-11-10
The given Differential equations is,
$$\displaystyle{y}{'''}+{10}{y}{''}+{25}{y}′={0}.$$
This equation can be re-written as,
$$\displaystyle{\left({D}^{{{3}}}+{10}{D}^{{{2}}}+{25}{D}\right)}{y}={0},$$ where $$\displaystyle{D}≡{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}.$$The auxiliary equation of (1) is, $$\displaystyle{D}^{{{3}}}+{10}{D}^{{{2}}}+{25}{D}={0}$$
$$\displaystyle\Rightarrow{D}{\left({D}^{{{2}}}+{10}{D}+{25}\right)}={0}$$
$$\displaystyle⟹{D}{\left({D}+{5}\right)}^{{{2}}}={0}$$
$$\displaystyle\Rightarrow{D}{\left({D}+{5}\right)}{\left({D}+{5}\right)}={0}$$
$$\displaystyle\Rightarrow{D}={0},-{5},-{5}.$$
Hence the required general solution is,
$$\displaystyle{y}={c}{1}{e}^{{{0}{x}}}+{\left({c}{2}+{c}{3}{x}\right)}{e}^{{-{5}{x}}}={c}{1}+{\left({c}{2}+{c}{3}{x}\right)}{e}^{{-{5}{x}}},{c}{1},{c}{2},{c}{3}$$ are arbitrary constants.

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