# (b) y''+y′+3⋅a5y=0

Question
Differential equations
$$\displaystyle{\left({b}\right)}{y}{''}+{y}′+{3}⋅{a}{5}{y}={0}$$

2021-03-12
The characteristic equation of this Differential equations is $$\displaystyle{r}^{{2}}+{r}+{3.25}={0}$$
Solve this quadratic equation: $$\displaystyle{r}=\frac{{-{1}\pm\sqrt{{{1}-{13}}}}}{{2}}=\frac{{-{1}\pm\sqrt{{12}}}}{{2}}=\frac{{-{1}\pm{2}{i}\sqrt{{3}}}}{{2}}=-{\left(\frac{{1}}{{2}}\right)}\pm{i}\sqrt{{3}}$$
Therefore, the solution is $$\displaystyle{y}={C}{1}{\left(\frac{{e}^{{-{{x}}}}}{{2}}\right)}{\cos{{\left({x}\sqrt{{3}}\right)}}}+{C}{2}{\left(\frac{{e}^{{-{{x}}}}}{{2}}\right)}{\sin{{\left({x}\sqrt{{3}}\right)}}}$$

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