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Question:

$y=e^{2 x}(a+b x)$

Solution:

$y=e^{2 x}(a+b x)$                …(1)

Differentiating both sides with respect to x, we get:

$y^{\prime}=2 e^{2 x}(a+b x)+e^{2 x} \cdot b$

$\Rightarrow y^{\prime}=e^{2 x}(2 a+2 b x+b)$       …(2)

Multiplying equation (1) with 2 and then subtracting it from equation (2), we get:

$y^{\prime}-2 y=e^{2 x}(2 a+2 b x+b)-e^{2 x}(2 a+2 b x)$

$\Rightarrow y^{\prime}-2 y=b e^{2 x}$                     …(3)

Differentiating both sides with respect to x, we get:

$y^{\prime \prime}-2 y^{\prime}=2 b e^{2 x}$        (4)

Dividing equation (4) by equation (3), we get:

$\frac{y^{\prime \prime}-2 y^{\prime}}{y^{\prime}-2 y}=2$

$\Rightarrow y^{\prime \prime}-2 y^{\prime}=2 y^{\prime}-4 y$

$\Rightarrow y^{\prime \prime}-4 y^{\prime}+4 y=0$

This is the required differential equation of the given curve.

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