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# Find a general solution. Check your answer by substitution ODEs of this kind have important applications to be discussed in Secs. 2.4, 2.7, and 2.9. 4y''-25y=0

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${\color{#c34632} {4y''-25y=0}}\;\;\;\; \Rightarrow \;\; \textbf{Homogeneous 2nd Order D.E , With Constant Coeff.}$

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

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

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

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

$\textbf{Then Characterestic Eq :}\;\; \Rightarrow \;\;{\color{#c34632} {4m^{2}-25=0}}$

$(2m-5)(2m+5)=0$

$m_{1}=\dfrac{5}{2}\;\;\;,\;\;\;m_{2}=\dfrac{-5}{2}$

$\therefore \;\; \textbf{Solution is : }\; {\color{#c34632} {y=C_{1}e^{m_{1}x}+C_{2}e^{m_{2}x}}}$

$\therefore \boxed{\;\color{#4257b2} {y=C_{1}e^{(5/2)x}+C_{2}e^{(-5/2)x}}\;}$

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