# solve the Differential equations y'''+10y''+25y'=0

solve the Differential equations $y+10y+25y=0$
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doplovif
The given Differential equations is,
$y{}^{‴}+10y{}^{″}+25y\prime =0.$
This equation can be re-written as,
$\left({D}^{3}+10{D}^{2}+25D\right)y=0,$ where $D\equiv \frac{d}{dx}.$The auxiliary equation of (1) is,${D}^{3}+10{D}^{2}+25D=0$
$⇒D\left({D}^{2}+10D+25\right)=0$
$⟹D{\left(D+5\right)}^{2}=0$
$⇒D\left(D+5\right)\left(D+5\right)=0$
$⇒D=0,-5,-5.$
Hence the required general solution is,
$y=c1{e}^{0x}+\left(c2+c3x\right){e}^{-5x}=c1+\left(c2+c3x\right){e}^{-5x},c1,c2,c3$ are arbitrary constants.