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Question

(D) Given equation: $y = a e^{2x} + b x e^{2x} = e^{2x}(a + bx)$.First derivative with respect to $x$: $\frac{dy}{dx} = 2e^{2x}(a + bx) + b e^{2x} = 2

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

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(D) Given equation: $y = a e^{2x} + b x e^{2x} = e^{2x}(a + bx)$.First derivative with respect to $x$: $\frac{dy}{dx} = 2e^{2x}(a + bx) + b e^{2x} = 2y + b e^{2x}$.Rearranging gives $b e^{2x} = \frac{dy}{dx} - 2y$ ... $(i)$.Second derivative with respect to $x$: $\frac{d^2y}{dx^2} = 2\frac{dy}{dx} + 2b e^{2x}$ ... $(ii)$.Substitute $(i)$ into $(ii)$:$\frac{d^2y}{dx^2} = 2\frac{dy}{dx} + 2(\frac{dy}{dx} - 2y)$.$\frac{d^2y}{dx^2} = 2\frac{dy}{dx} + 2\frac{dy}{dx} - 4y$.$\frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 4y = 0$.Thus,the differential equation is $y^{\prime \prime} - 4y^{\prime} + 4y = 0$.

The differential equation formed by eliminating $a$ and $b$ from the equation $y=a e^{2 x}+b x e^{2 x}$ is

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