If c_1, c_2, c_3, c_4, c_5 are arbitrary cons | vedclass

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(A) The given general solution is $y=(c_1+c_2) \sin (x+c_3)+c_4 e^{x+c_5}$.We can simplify the expression by substituting constants:Let $A = c_1+c_2$

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$3$

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(A) The given general solution is $y=(c_1+c_2) \sin (x+c_3)+c_4 e^{x+c_5}$.We can simplify the expression by substituting constants:Let $A = c_1+c_2$ and $B = c_4 e^{c_5}$.Then 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 $K_3 = B$.Thus,$y = K_1 \sin x + K_2 \cos x + K_3 e^x$.There are $3$ essential arbitrary constants $(K_1, K_2, K_3)$.The order of a differential equation is equal to the number of essential arbitrary constants in its general solution.Therefore,the order of the differential equation is $3$.

If $c_1, c_2, c_3, c_4, c_5$ are arbitrary constants,then the order of the differential equation whose general solution is $y=(c_1+c_2) \sin (x+c_3)+c_4 e^{x+c_5}$ is

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