Find differential equation that satisfy all values of the arbitrary constants :

$\displaystyle y = Ax^2 + e^x$

The answer is

$\displaystyle x\frac{dy}{dx} = 2y + (x - 2)e^x $

Right, so i start off by eliminating e^x

$\displaystyle Equation one : y = Ax^2 + e^x$

$\displaystyle Equation two : \frac{dy}{dx} = 2Ax + e^x$

$\displaystyle Equation three: \frac{d^2y}{dx^2} = 2A + e^x$

Eliminate $\displaystyle e^x$

Minus both equations 1 & 2 and i get

$\displaystyle y - \frac{dy}{dx} = A(x^2 - 2x)$

Minus both equations 2 & 3 and i get

$\displaystyle \frac{d^2y}{dx^2} - \frac{dy}{dx} = A(2 - 2x)$

Now eliminate A

$\displaystyle \frac{y - \frac{dy}{dx}}{(x^2 - 2x)} - \frac{\frac{d^2y}{dx^2} - \frac{dy}{dx}}{(2 - 2x)} = 0$

SO then i cross multiply and get the wrong answer...am i doing something wrong above?

Thank-you