Find the differential equation by eliminating arbitrary constants… - Vidyadip

Question

Find the differential equation by eliminating arbitrary constants from the relation $y = (c_1 + c_2x)e^x$

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Answer

$y=\left(c_1+c_2 x\right) e^x\ldots(i)$
Here, $c_1$ and $c_2$ are arbitrary constants.
Differentiating w.r.t. $x$, we get
$ \frac{ d y}{ d x}=\left( c _1+ c _2 x \right) e ^{ x }+ c _2 e ^{ x }$
$\therefore \frac{ d y}{ d x}= y + c _2 e ^{ x }\ldots(ii)\ldots[From(i)] $
Again, differentiating w.r.t. $x$, we get
$ \frac{ d ^2 y}{ d x^2}=\frac{ d y}{ d x}+ c _2 e ^x$
$\therefore c _2 e ^{ x }=\frac{ d ^2 y}{ d x^2}-\frac{ d y}{ d x}\ldots(iii) $
Substituting (iii) in (ii), we get
$ \frac{ d y}{ d x}=y+\frac{ d ^2 y}{ d x^2}-\frac{ d y}{ d x}$
$\therefore \frac{ d ^2 y}{ d x^2}-2 \frac{ d y}{ d x}+y=0 $

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