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What is a general solution to the differential equation #dy/dx=xe^-y#?

Answer 1

Scarlett Allison

This is a separable differential equation.

#dy/(dx) = xe^(-y)#

#dy/(dx) = x*1/(e^y) #

#e^y * dy/(dx) = x#

#e^y dy = x dx#

#int e^y dy = int x dx#

#e^y = 1/2 x^2 + C#

#y = ln(1/2 x^2+C)#

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Answer 2

Ezra Adams

The general solution to the differential equation ( \frac{dy}{dx} = xe^{-y} ) is given implicitly by ( y = -\ln(C - \frac{x^2}{2}) ), where ( C ) is an arbitrary constant.

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Answer 3

Elijah Abram

The general solution to the differential equation (\frac{dy}{dx} = x e^{-y}) is given implicitly by:

[ y = -\ln(C - \frac{x^2}{2}) ]

where ( C ) is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.