Question

Obtain the differential equation by eliminating the arbitrary constants from the given equation, $y=( c_{1}+c_{2}x)\ e^{x}$

Answer 1

To obtain the differential equation by eliminating the arbitrary constants from the given equation, $y=( c_{1}+c_{2}x)\ e^{x}$

we must keep consideration that it is to be differentiate twice,since there are two arbitrary

$y = (c_{1} + c_{2} \: x) {e}^{x} \\ \\ y = c_{1} {e}^{x}+ c_{2} \: x {e}^{x}...eq1 \\ \\ \frac{dy}{dx} = c_{1} {e}^{x} + c_{2}(x {e}^{x} + {e}^{x} ) \\ \\ \frac{dy}{dx} = c_{1} {e}^{x}+ c_{2} \: x {e}^{x} + c_{2} {e}^{x} \\ \\ \frac{dy}{dx} = y + c_{2} {e}^{x} \: \: \: \: substitute \: from \: eq1 \\ \\ \frac{ {d}^{2}y }{ {dx}^{2} } = \frac{dy}{dx} + c_{2} \: {e}^{x} \\ \\ \frac{ {d}^{2}y }{ {dx}^{2} } = \frac{dy}{dx} + \frac{dy}{dx} - y \\ \\ \frac{ {d}^{2}y }{ {dx}^{2} } - 2 \frac{dy}{dx} + y = 0$
is the solution of given problem.