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The order of the differential equation w...

The order of the differential equation whose general solution is given by `y=(C_1+C_2)cos(x+C_3)-C_4e^(x+4_5),` where `C_1,C_2,C_3,C_4,C_5` , are arbitrary constants, is (a) 5 (b) 4 (c) 3 (d) 2

A

(a) 5

B

(b) 4

C

(c) 3

D

(d) 2

Text Solution

Verified by Experts

The correct Answer is:
(c)
Given, `y=(c_(1)+c_(2))cos (x+c_(3))-c_(4)e^(x+c_(5)) …. (i)`
` rArr y=(c_(1)+c_(2))cos (x+c_(3))-c_(4)e^(x)cdot e^(c_(5)) `
Now, let `c_(1)+c_(2)=A,c_(3)=B,c_(4)e^(c_(5))=c`
`rArr y=A cos (x+B)-ce^(x) …(ii)`
On differentiating w.r.t.x, we get
`dy/dx=-A sin (A+B)-ce^(x) …(iii)`
Again, on differentiating w.r.t.x, we get
`(d^(2)y)/dx^(2)=-Acos (x+B)-ce^(x) … (iv)`
`rArr (d^(2)y)/dx^(2)=-y-2ce^(x) … (v)`
`rArr (d^(2)y)/dx^(2)+y=-2ce^(x)`
Again, on differentiating w.r.t.x, we get
` (d^(3)y)/dx^(2)+dy/dx=-2ce^(x) ..(vi)`
`rArr (d^(3)y)/dx^(2)+dy/dx=(d^(2)y)/dx^(2)+y [from Eq. (v)]`
which is a differential equation of order 3.
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