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What is a solution to the differential equation #(x^2)(e^y)dy/dx=4#?

Answer 1

Charles Arocha

#y=ln(C-4/x)#

We can separate the variables:

#x^2e^ydy/dx=4" "=>" "e^ydy=4x^-2dx#

Integrate both sides:

#inte^ydy=4intx^-2dx" "=>" "e^y=-4/x+C" "=>" "y=ln(C-4/x)#

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Answer 2

Aurora Alvarado

To solve the given differential equation ((x^2)(e^y)\frac{dy}{dx}=4), you can rearrange it and integrate both sides with respect to (x). First, divide both sides by (x^2e^y), then integrate with respect to (x).

[\frac{dy}{dx} = \frac{4}{x^2e^y}]

[e^y dy = \frac{4}{x^2} dx]

Now, integrate both sides:

[\int e^y ,dy = \int \frac{4}{x^2} ,dx]

[e^y = -\frac{4}{x} + C]

Where (C) is the constant of integration. This is the general solution to the given differential equation.

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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.