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${\color{#c34632} {100y''+240y'+(196\pi^{2} +144)y=0}}\;\;\;\; \Rightarrow \;\; \textbf{Homogeneous 2nd Order D.E , With Constant Coeff.}$

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

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

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

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

$\textbf{Then Characterestic Eq :}\;\; \Rightarrow \;\;{\color{#c34632} {100m^{2}+240m+(196\pi^{2} +144)=0}}$

$m_{1,2}=\dfrac{-b \;\pm \sqrt{b^{2}-4ac}}{2a}=\dfrac{-240\;\pm \sqrt{57600-(78400\pi^{2} +57600)}}{200}$

$m_{1,2}=\dfrac{-240\;\pm 280\pi i}{200}=\dfrac{-240-280\pi i}{200}\;\;,\;\; \dfrac{-240+280\pi i}{200}$

$m_{1,2}=\dfrac{-6}{5}\pm \dfrac{7}{5}\pi i\;\;\;\; \Rightarrow \;\; \boxed{\;m_{1,2}=\alpha \pm \beta i\;}$

$\boxed{\;\alpha =\dfrac{-6}{5}\;}\;\;\;.\;\;\; \boxed{\;\beta =\dfrac{7}{5}\pi \;}$

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

$\therefore \boxed{\;\color{#4257b2} {y=e^{\dfrac{-6}{5} x}(C_{1}\cos(\dfrac{7}{5}\pi x)+C_{2}\sin(\dfrac{7}{5}\pi x))}\;}$

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