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

Answer 1

Cameron Alexander

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

#dy/dx=(1+x)(1+y)#

this is separable!

#1/(1+y) dy/dx=1+x#

so we integrate both sides

#int \ 1/(1+y) dy/dx \ dx=int \ 1+x \ dx#

#int \ 1/(1+y) \ dy=int \ 1+x \ dx#

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

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

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

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Answer 2

Adam Adkins

The solution to the differential equation ( \frac{dy}{dx} = (1+x)(1+y) ) is given implicitly by:

[ y = -1 - x + \sqrt{C(1+x)} ]

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.