GATE2006: Higher Order Differential Equation

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Question: GATE2006(2 Marks)

For \(\displaystyle \frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 3y = 3e^{2x} \), the particular integral is 

\(\displaystyle (A) \frac{1}{15} e^{2x} \)                   \(\displaystyle (B) \frac{1}{5} e^{2x} \)
\((C) 3e^{2x} \) \((D) C_1e^{-x} + C_2e^{-3x} \)

Solution:

(D2 + 4D + 3)y = 3e2x

Particular Integral \(\displaystyle P.I = \frac{3e^{2x}}{D^2+4D+3} \\ = \displaystyle \frac{3e^{2x}}{(D+1)(D+3)} \)

\(\displaystyle P.I \text { for }  =\frac{e^{ax}}{f(D)} = \frac{e^{ax}}{f(a)} \)

In this case, a = 2

\(\displaystyle P.I = \frac{3e^{2x}}{(2+1)(2+3)} = \frac{1}{5} e^{2x} \)

Answer: (B)