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(B) The given solution is $y = a e^{2x} + b x e^{2x} + C$ ...$(i)$Here,$a$ and $b$ are arbitrary constants,while $C$ is a fixed constant.The order of

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$2$

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(B) The given solution is $y = a e^{2x} + b x e^{2x} + C$ ...$(i)$Here,$a$ and $b$ are arbitrary constants,while $C$ is a fixed constant.The order of a differential equation is equal to the number of arbitrary constants present in its general solution.Since there are two arbitrary constants ($a$ and $b$),the order of the differential equation is $2$.To verify,we differentiate $y$ with respect to $x$:$y_1 = 2a e^{2x} + b(e^{2x} + 2x e^{2x}) = (2a + b)e^{2x} + 2bx e^{2x}$ ...(ii)$y_2 = 2(2a + b)e^{2x} + 2b(e^{2x} + 2x e^{2x}) = (4a + 4b)e^{2x} + 4bx e^{2x}$ ...(iii)By eliminating $a$ and $b$ from these equations,we obtain a second-order differential equation.

Let $a$ and $b$ be arbitrary constants and $C$ be a fixed constant. If $y = a e^{2x} + b x e^{2x} + C$ is the general solution of a differential equation,then the order of that differential equation is

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