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Find a general solution. Check your answer by substitution. y''+y'+3.25y=0

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

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

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

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

y        m2      ,      y        m      ,      y        1\boxed{y'' \;\; \Rightarrow \;\; m^{2}\;\;\;,\;\;\; y'\;\; \Rightarrow \;\; m\;\;\;,\;\;\;y\;\; \Rightarrow \;\; 1}

Then Characterestic Eq :        m2+m+3.25=0\textbf{Then Characterestic Eq :}\;\; \Rightarrow \;\;{\color{#c34632} {m^{2}+m+3.25=0}}

m1,2=b  ±b24ac2a=1  ±1132m_{1,2}=\dfrac{-b \;\pm \sqrt{b^{2}-4ac}}{2a}=\dfrac{-1\;\pm \sqrt{1-13}}{2}

m1,2=1  ±23i2m_{1,2}=\dfrac{-1\;\pm 2\sqrt{3}i}{2}

m1,2=12±3  i              m=α±βi  m_{1,2}=\dfrac{-1}{2}\pm \sqrt{3}\;i\;\;\;\; \Rightarrow \;\; \boxed{\;m=\alpha \pm \beta i\;}

α=12      ,      β=3\therefore \alpha = \dfrac{-1}{2} \;\;\; , \;\;\; \beta =\sqrt{3}

    Solution is :   y=eαx[C1cosβ+C2sinβ]          For Complix Roots\therefore \;\; \textbf{Solution is : }\; \boxed{\color{#c34632} {y=e^{\alpha x}[C_{1}\cos \beta +C_{2}\sin \beta ]}}\;\;\; \Rightarrow \;\; \textbf{For Complix Roots}

  y=e(1/2)x[C1cos3+C2sin3]  \therefore \boxed{\;\color{#4257b2} {y=e^{(-1/2)x}[C_{1}\cos \sqrt{3} +C_{2}\sin \sqrt{3} ]}\;}

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