What is a particular solution to the differential equation dydx=ex−y with y(0)=2?

Explanation:this is separabledydx=ex−y=exe−yeydydx=ex∫eydydxdx=∫exdx∫eydy=∫exdxey=ex+Cy(0)=2⇒e2=1+C⇒C=e2−1ey=ex+e2−1y=ln⁡(ex+e2−1)

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gabolzm6d

2022-04-17

What is a particular solution to the differential equation

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

with y(0)=2?

folytonr4qx

Beginner2022-04-18Added 14 answers

Explanation:
this is separable

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

e^{y} \frac{d y}{d x} = e^{x}

\int e^{y} \frac{d y}{d x} d x = \int e^{x} d x

\int e^{y} d y = \int e^{x} d x

e^{y} = e^{x} + C

y (0) = 2 \Rightarrow e^{2} = 1 + C \Rightarrow C = e^{2} - 1

e^{y} = e^{x} + e^{2} - 1

y = \ln (e^{x} + e^{2} - 1)

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