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What is a solution to the differential equation #y'=x/y=x/(1+y)#?

Answer 1

Ezra Adams

as currently stated, that's not a DE

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Answer 2

William Abdalla

To find a solution to the differential equation ( y' = \frac{x}{y} = \frac{x}{1+y} ), we can separate variables and integrate.

Separate variables: ( \frac{dy}{dx} = \frac{x}{y} ) ( y , dy = x , dx )
Integrate both sides: ( \int y , dy = \int x , dx ) ( \frac{y^2}{2} = \frac{x^2}{2} + C )
Solve for ( y ): ( y^2 = x^2 + C ) ( y = \sqrt{x^2 + C} )
This equation represents 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.