For the given equation, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
$y = e^{2 x} (a + b x)$

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Solution

Given, $y = e^{2 x} (a + b x)$ ....(i)
On differentiating both sides w.r.t. x, we get
$\frac{d y}{d x} = e^{2 x} \frac{d}{d x} (a + b x) + (a + b x) \frac{d}{d x} e^{2 x}$ (using product rule of differentiation)
$\Rightarrow \frac{d y}{d x} = e^{2 x} (b) + 2 e^{2 x} (a + b x) \Rightarrow y^{'} = 2 y + b e^{2 x}$ ....(ii)
Again differentiating Eq. (ii) w.r.t. x, we get
$y^{''} = 2 y^{'} + 2 b e^{2 x}$ ....(iii)
On multiplying Eq.(ii) by 2 and then substracting from Eq. (iii), we get
$y^{''} = 2 y^{'} + 2 (y^{'} - 2 y) \Rightarrow y^{''} = 2 y^{'} + 2 y^{'} - 4 y \Rightarrow y^{''} - 4 y^{'} + 4 y = 0$
which is the required differential equation.

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For the given equation, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.​ y=e2x(a+bx)

For the given equation, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
$y = e^{2 x} (a + b x)$