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(D) The given general solution is $y = a_1(a_2 + a_3) \cdot \cos(x + a_4) - a_5 e^{x + a_6}$.Let $C_1 = a_1(a_2 + a_3)$,$C_2 = a_4$,and $C_3 = a_5$,$C

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$3$

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(D) The given general solution is $y = a_1(a_2 + a_3) \cdot \cos(x + a_4) - a_5 e^{x + a_6}$.Let $C_1 = a_1(a_2 + a_3)$,$C_2 = a_4$,and $C_3 = a_5$,$C_4 = a_6$.Then the equation can be rewritten as $y = C_1 \cos(x + C_2) - C_3 e^{x + C_4}$.Using the trigonometric identity $\cos(x + C_2) = \cos x \cos C_2 - \sin x \sin C_2$,we get:$y = C_1(\cos x \cos C_2 - \sin x \sin C_2) - C_3 e^{x + C_4}$.$y = (C_1 \cos C_2) \cos x - (C_1 \sin C_2) \sin x - (C_3 e^{C_4}) e^x$.Let $A = C_1 \cos C_2$,$B = -C_1 \sin C_2$,and $D = -C_3 e^{C_4}$.Thus,the equation simplifies to $y = A \cos x + B \sin x + D e^x$.There are $3$ independent arbitrary constants $(A, B, D)$.The order of a differential equation is equal to the number of independent arbitrary constants in its general solution.Therefore,the order of the differential equation is $3$.

The order of the differential equation whose general solution is $y = a_1(a_2 + a_3) \cdot \cos(x + a_4) - a_5 e^{x + a_6}$ is . . . . . . .

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