We #160;have,dydx+2y=e3x #160; #160; #160; #160; #160; #160; #160; #160; #160; #160; #160;.....(1)Clearly, #160;it #160;is #160;a ...

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Asymptotes

dydx +2y=e3x

Question

$\frac{d y}{d x} + 2 y = e^{3 x}$

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Solution

$We have, \frac{d y}{d x} + 2 y = e^{3 x} . . . . . (1) Clearly, it is a linear differential equation of the form \frac{d y}{d x} + P y = Q where P = 2 Q = e^{3 x} ∴ I . F . = e^{\int P d x} = e^{\int 2 d x} = e^{2 x} Multiplying both sides of (1) by e^{2 x}, we get e^{2 x} (\frac{d y}{d x} + 2 y) = e^{2 x} e^{3 x} \Rightarrow e^{2 x} \frac{d y}{d x} + 2 e^{2 x} y = e^{5 x} Integrating both sides with respect to x, we get y e^{2 x} = \int e^{5 x} d x + C \Rightarrow y e^{2 x} = \frac{e^{5 x}}{5} + C \Rightarrow y = \frac{1}{5} e^{3 x} + C e^{- 2 x} Hence, y = \frac{1}{5} e^{3 x} + C e^{- 2 x} is the required solution .$

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