# How do you find the general solution to #dy/dx=xe^y#?

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To find the general solution to the differential equation (\frac{dy}{dx} = x e^y), we can separate variables and integrate both sides with respect to (x) and (y) respectively. Here's how:

[ \frac{dy}{dx} = x e^y ]

Separate variables:

[ \frac{dy}{e^y} = x , dx ]

Integrate both sides:

[ \int \frac{1}{e^y} , dy = \int x , dx ]

The integral of (1/e^y) can be evaluated as (-e^{-y}), and the integral of (x) with respect to (x) is (\frac{1}{2}x^2):

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

where (C) is the constant of integration.

To find (y), we can solve for (y):

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

Taking the reciprocal of both sides:

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

[ e^y = \frac{1}{\frac{1}{2}x^2 - C} ]

[ y = \ln\left(\frac{1}{\frac{1}{2}x^2 - C}\right) ]

So, the general solution to the differential equation (\frac{dy}{dx} = x e^y) is (y = \ln\left(\frac{1}{\frac{1}{2}x^2 - C}\right)), where (C) is an arbitrary constant.

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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.

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.

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.

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