# Find the general solution of the given second-order

Find the general solution of the given second-order differential equations
$2y{}^{″}+2{y}^{\prime }+y=0$
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Given differentia equation, $2y{}^{″}+2{y}^{\prime }+y=0$
Auxilliary equation
$2{m}^{2}+2m+1=0$
$m=\frac{-2±\sqrt{{2}^{2}-4\left(2\right)\left(1\right)}}{2\left(2\right)}$
$m=\frac{-2±\sqrt{-4}}{4}$
$m=-\frac{1}{2}±\frac{1}{2}i$
So, the general solution:$y={c}_{1}\mathrm{cos}\left(\frac{1}{2}x\right)+{c}_{2}\mathrm{sin}\left(\frac{1}{2}x\right){e}^{-\frac{1}{2}x}$
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