Question

Obtain the differential equation by eliminating the arbitrary constants from the given equation, $y=c_{1}e^{3x} +c_{2}e^{2x}$

Answer 1

To obtain the differential equation by eliminating the arbitrary constants from the given equation,
$y=c_{1}e^{3x} +c_{2}e^{2x}$,
differentiate the equation twice,and eliminate the constants

$y=c_{1}e^{3x} +c_{2}e^{2x} \\ \\ \frac{dy}{dx} = 3c_{1} {e}^{3x} + 2c_{2} {e}^{2x} ...eq1 \\ \\ \frac{ {d}^{2}y }{ {dx}^{2} } = 9c_{1} {e}^{3x} + 6c_{2} {e}^{2x} \\ \\ = 3(3c_{1} {e}^{3x} + 2c_{2} {e}^{2x})...eq2 \\ \\ substitute \: from \: eq1 \: in \: eq2 \\ \\ \frac{ {d}^{2}y }{ {dx}^{2} } = 3 \frac{dy}{dx} \\ \\ \frac{ {d}^{2}y }{ {dx}^{2} } - 3 \frac{dy}{dx} = 0$

is the required differential equation.