Question

Form the differential equation representing the family of curves $y=e^{2x}(a+bx),$ where $a\text{and}b$ are arbitrary constants.

Answer 1

$y=e^{2x}(a+bx)$
It consists of $2$ arbitrary constants.
This means we can differentiate it twice to get the differential equation.
Multiply by $e^{-2x}$ on both sides, we get
$e^{-2x}y=a+bx$
Differentiating wrt $x,$ we get
$e^{-2x}\frac{dy}{dx}+(-2)e^{-2x}y=b\Rightarrow e^{-2x}(\frac{dy}{dx}-2y)=b$
Again, differentiating wrt $x,$ we get
$e^{-2x}(\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx})+(-2)e^{-2x}(\frac{dy}{dx}-2y)=0\Rightarrow e^{-2x}(\frac{d^{2}y}{d{x}^2}-4\frac{dy}{dx}+4y)=0\Rightarrow \frac{d^{2}y}{d{x}^2}-4\frac{dy}{dx}+4y=0[\because e^{-2x}\ne 0]$
This is the required differential equation.