GATE2006: Higher Order Differential Equation

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

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