Let c_1, c_2, c_3, c_4 be arbitrary constants | vedclass

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(C) Given the equation: $y=c_1 e^x+c_2 e^{\log _{e} x}+c_3 \sin ^2 x-c_4\left(\cos ^2 x-1\right)$Using the property $e^{\log _{e} x} = x$ and $\cos^2

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$3$

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(C) Given the equation: $y=c_1 e^x+c_2 e^{\log _{e} x}+c_3 \sin ^2 x-c_4\left(\cos ^2 x-1\right)$Using the property $e^{\log _{e} x} = x$ and $\cos^2 x - 1 = -\sin^2 x$,we simplify:$y = c_1 e^x + c_2 x + c_3 \sin^2 x - c_4(-\sin^2 x)$$y = c_1 e^x + c_2 x + (c_3 + c_4) \sin^2 x$Let $C = c_3 + c_4$. Then the equation becomes:$y = c_1 e^x + c_2 x + C \sin^2 x$This equation contains $3$ independent arbitrary constants $(c_1, c_2, C)$.The order of a differential equation is equal to the number of independent arbitrary constants present in its general solution.Since there are $3$ independent constants,the order of the differential equation is $3$.

Let $c_1, c_2, c_3, c_4$ be arbitrary constants. The order of the differential equation,corresponding to $y=c_1 e^x+c_2 e^{\log _{e} x}+c_3 \sin ^2 x-c_4\left(\cos ^2 x-1\right)$ is

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