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Asymptotes

dydx +2 y=6 e...

Question

$\frac{d y}{d x} + 2 y = 6 e^{x}$

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Solution

$We have, \frac{d y}{d x} + 2 y = 6 e^{x} . . . . . (1) Clearly, it is a linear differential equation of the form \frac{d y}{d x} + P y = Q where P = 2 Q = 6 e^{x} ∴ I . F . = e^{\int P d x} = e^{\int 2 d x} = e^{2 x} Multiplying both sides of (1) by e^{2 x}, we get e^{2 x} (\frac{d y}{d x} + 2 y) = 6 e^{2 x} e^{x} \Rightarrow e^{2 x} \frac{d y}{d x} + 2 e^{2 x} y = 6 e^{3 x} Integrating both sides with respect to x, we get y e^{2 x} = 6 \int e^{3 x} d x + C \Rightarrow y e^{2 x} = 6 \frac{e^{3 x}}{3} + C \Rightarrow y e^{2 x} = 2 e^{3 x} + C Hence, y e^{2 x} = 2 e^{3 x} + C is the required solution .$

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