GATE1996: Higher Order Differential Equation

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Question: GATE1996 (2 Marks)

The particular solution for the differential equation \(\displaystyle \frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 5 cos x \) is

(a) 0.5 cos x + 1.5 sin x                     (b) 1.5 cosx + 0.5 sin x 
(c) 1.5 sin x (d) 0.5 cos x

Solution:

(D2 + 3D + 2) y = 5cosx

Particular Integral: \(\displaystyle P.I =\frac{5 cosx}{D^2 + 3D +2} \)

  \(\displaystyle \frac{cosax}{f(D^2)} = \frac{cosax}{f(-a^2)} \)

In this case, a = 1; D2 = –1

 \(\displaystyle \frac{5cosx}{3D + 1} \)

Multiply and divide by (1 – 3D)

 \(\displaystyle \frac{5cosx}{1+3D} \times \frac{1-3D}{1-3D} \\ \displaystyle = \frac{(1-3D) 5cosx}{1-9D^2} \)

In this case, a = 1; D2 = –1

\(\displaystyle \frac{(1-3D)5cos x}{1-9D^2} = \frac{(1-3D)5cosx}{10} \\ \displaystyle = \frac{(1-3D)5cosx}{10} \)

\(\displaystyle \frac{5cosx + 15 sinx}{10} = 0.5 cosx+1.5 sin x \)

Answer: (A)