The differential equation whose general solution is given by `y=(c_1cos(x+c_2))-(c_3e^((-x+c_4)))+(c_5sinx),` where `c_1,c_2,c_3,c_4,c_5` are arbitrary constants, is

Options

A. `((d^4y)/(dx^4))- ((d^2y)/(dx^2)) + y = 0`
B. `((d^4y)/(dx^4))+ ((d^2y)/(dx^2)) +((dy)/(dx))+ y = 0`
C. `((d^5y)/(dx^5)) + y = 0`
D. `((d^4y)/(dx^4))- ((d^2y)/(dx^2)) +((dy)/(dx))+ y = 0`

Answer

b

Explanation

Given , `y = c_(1) cos ( x+c_(2))+c_(3) sin (x+c_(4)) +c_(5)e^(x) +c_(6)`
` y = c_(1) [ cos x cos c_(2) - sin x sin c_(2)]`
` + c_(3) [ sin x cos_(4) + cos x sin c_(4)] + c_(5) e^(x) +c_(6)`
` = cos x (c_(1) cos c_(2) + c_(3) sin c_(4))`
` + sin x ( - c_(1) sin c_(2) +c_(3) cos c_(4)) + c_(5)e^(x) +c_(6)`
` = A cos x + B sin x + Ce^(x) + D`
where ` A = c_(1) cos c_(2) + c_(3) sin c_(4)`
`B = - c_(1) sin c_(2) + c_(3) cos c_(4) , C= c_(5) , D = c_(6)`
Hence , order is 4 .