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(D) The given general solution is $y = (C_1 + C_2) \sin (x + C_3) - C_4 e^{x + C_5}$.We can simplify the constants as follows:Let $A = (C_1 + C_2)$. S

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$3$

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(D) The given general solution is $y = (C_1 + C_2) \sin (x + C_3) - C_4 e^{x + C_5}$.We can simplify the constants as follows:Let $A = (C_1 + C_2)$. Since $C_1$ and $C_2$ are arbitrary constants,their sum $A$ is also an arbitrary constant.Let $B = C_4 e^{C_5}$. Since $C_4$ and $C_5$ are arbitrary constants,$B$ is also an arbitrary constant.Now,the equation becomes $y = A \sin (x + C_3) - B e^x$.Using the trigonometric identity $\sin (x + C_3) = \sin x \cos C_3 + \cos x \sin C_3$,we get:$y = A (\sin x \cos C_3 + \cos x \sin C_3) - B e^x$$y = (A \cos C_3) \sin x + (A \sin C_3) \cos x - B e^x$.Let $K_1 = A \cos C_3$,$K_2 = A \sin C_3$,and $K_3 = -B$.These $K_1, K_2, K_3$ are independent arbitrary constants.Thus,the equation simplifies to $y = K_1 \sin x + K_2 \cos x + K_3 e^x$.Since there are $3$ independent arbitrary constants in the general solution,the order of the corresponding differential equation is $3$.

The order of the differential equation whose general solution is given by $y = (C_1 + C_2) \sin (x + C_3) - C_4 e^{x + C_5}$ is (where $C_1, C_2, C_3, C_4, C_5$ are arbitrary constants).

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