Solve d y/d x+y=e^{-3 x}

Correct Answer is : y=12e3x+Cexy = 1/2 e³x} + C e^{x}y=21e3x+Cex

Correct Answer is : y=12e3x+Cexy = 1/2 e³x} + C e^{x}y=21e3x+Cex

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Army Group X Practice Set 2

Question : 56 of 70

Marks: +1, -0

Solve

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

Solution:

Given equation,

\frac{d y}{d x}+y=e^{-3 x},

it is a linear differential equation.

Comparing the given equation with

\frac{d y}{d x}+P y=Q,

we get

P=1,

Q=e^{-3 x}

\therefore \mid F=e^{\int P\,dx}=e^{\int 1\cdot dx}=e^{x}

Solution,

y \cdot \text{IF} = \int Q \cdot ( , ) \, dx + C

\Rightarrow y e^{x} = \int e^{-3x} \cdot e^{x} \, dx + C

\Rightarrow y \cdot e^{x} = \int e^{-2x} \, dx + C

\Rightarrow y e^{x} \& = \frac{e^{-2x}}{-2} + C

\Rightarrow y \& = -\frac{1}{2} e^{-3x} + C e^{-x}

y = -\frac{1}{2} e^{-3x} + C e^{-x}

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