Differential EquationHard
Question
The order of the differential equations whose general solution is given by y = (c1 + c2) cos(x + c3) - c4ex+c5 where c1, c2, c3, c4, c5, are arbitrary constants, is
Options
A.5
B.4
C.3
D.2
Solution
Given, y = (c1 + c2) cos (x + c2) - c4ex + c5 .....(i)
⇒ y = (c1 + c2) cos (x + c2) - c4ex . ec5
Now, let c1 + c2 = A,c3 = B, c4ec5 = c
⇒ y = Acos(x + B) - cex .....(ii)
On differentiating w. r. t. x, we get
- A sin (A + B) - cex .....(iii)
Again differentiating w. r. t. x, we get
- A cos (x + B) - cex .....(iv)
⇒- y - 2cex .....(v)
⇒
+ y = - 2cex
Again differentiating w. r. t. x, we get
- 2cex .....(vi)
⇒
[from Eq. (v)]
Which is a differential equation of order 3.is :
or we can write this equation as
y = (k1cos (x + k2) - k3ex
so there are three arbitrary constant so the general solution will be of third degree.
⇒ y = (c1 + c2) cos (x + c2) - c4ex . ec5
Now, let c1 + c2 = A,c3 = B, c4ec5 = c
⇒ y = Acos(x + B) - cex .....(ii)
On differentiating w. r. t. x, we get
- A sin (A + B) - cex .....(iii) Again differentiating w. r. t. x, we get
- A cos (x + B) - cex .....(iv)⇒
- y - 2cex .....(v)⇒
+ y = - 2cex Again differentiating w. r. t. x, we get
- 2cex .....(vi)⇒
[from Eq. (v)]Which is a differential equation of order 3.is :
or we can write this equation as
y = (k1cos (x + k2) - k3ex
so there are three arbitrary constant so the general solution will be of third degree.
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