Obtain the differential equation by eliminating arbitrary constan… - Vidyadip

Question

Obtain the differential equation by eliminating arbitrary constants from the following equations : $y=\left(c_1+c_2 x\right) e^x$

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Answer

$
\begin{aligned}
& y=\left(c_1+c_2 x\right) e^x \\
& \therefore e^{-x} y=c_1+c_2 x
\end{aligned}
$
Differentiating w.r.t. $x$, we get
$
e^{-x} \cdot \frac{d y}{d x}+y \cdot e^{-x}(-1)=0+c_2 \times 1
$
$
\therefore e^{-x}\left(\frac{d y}{d x}-y\right)=c_2
$
Differentiating again w.r.t. $x$, we get
$
\begin{aligned}
& e^{-x}\left(\frac{d^2 y}{d x^2}-\frac{d y}{d x}\right)+\left(\frac{d y}{d x}-y\right) \cdot e^{-x}(-1)=0 \\
& \therefore e^{-x}\left(\frac{d^2 y}{d x^2}-\frac{d y}{d x}-\frac{d y}{d x}+y\right)=0 \\
& \therefore \frac{d^2 y}{d x^2}-2 \frac{d y}{d x}+y=0
\end{aligned}
$
This is the required D.E.

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