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${\color{#c34632} {y''+y'+3.25y=0}}\;\;\;\; \Rightarrow \;\; \textbf{Homogeneous 2nd Order D.E , With Constant Coeff.}$

$\Rightarrow \textbf{Form :}\;\; \boxed{\;ay''+by'+cy=0\;}$

$Where\; : \;a\;,\;b\;,\;c \;\; \Rightarrow Constants$

$In\; order\; to\; solve\; this\; D.E \;, \;we\; need \;to\; get\; the\; "characterestic\; equation\;" by\; replacing$

$\boxed{y'' \;\; \Rightarrow \;\; m^{2}\;\;\;,\;\;\; y'\;\; \Rightarrow \;\; m\;\;\;,\;\;\;y\;\; \Rightarrow \;\; 1}$

$\textbf{Then Characterestic Eq :}\;\; \Rightarrow \;\;{\color{#c34632} {m^{2}+m+3.25=0}}$

$m_{1,2}=\dfrac{-b \;\pm \sqrt{b^{2}-4ac}}{2a}=\dfrac{-1\;\pm \sqrt{1-13}}{2}$

$m_{1,2}=\dfrac{-1\;\pm 2\sqrt{3}i}{2}$

$m_{1,2}=\dfrac{-1}{2}\pm \sqrt{3}\;i\;\;\;\; \Rightarrow \;\; \boxed{\;m=\alpha \pm \beta i\;}$

$\therefore \alpha = \dfrac{-1}{2} \;\;\; , \;\;\; \beta =\sqrt{3}$

$\therefore \;\; \textbf{Solution is : }\; \boxed{\color{#c34632} {y=e^{\alpha x}[C_{1}\cos \beta +C_{2}\sin \beta ]}}\;\;\; \Rightarrow \;\; \textbf{For Complix Roots}$

$\therefore \boxed{\;\color{#4257b2} {y=e^{(-1/2)x}[C_{1}\cos \sqrt{3} +C_{2}\sin \sqrt{3} ]}\;}$

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