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What is a general solution to the differential equation #y'=2+2x^2+y+x^2y#?

Answer 1

Isabella Ackerman

#y = Ce^(x + x^3/3) - 2#

#y'=2+2x^2+y+x^2y#

this is separable,

#y'=2(1+x^2)+y(1+x^2)#

#y'=(2 + y)(1+x^2)#

#1/(2+y) y'=(1+x^2)#

#int \ 1/(2+y) y' \ dx =int \ (1+x^2) \ dx#

#int \ d/dx( ln(2+y) ) \ dx =int \ (1+x^2) \ dx#

#ln(2+y) =x + x^3/3 + C#

#2+y = e^(x + x^3/3 + C)#

#2+y = Ce^(x + x^3/3) #

#y = Ce^(x + x^3/3) - 2#

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Answer 2

Luke Alvarez

The general solution to the differential equation ( y' = 2 + 2x^2 + y + x^2y ) is given by:

[ y(x) = -x^2 - 2x - 2 + Ce^{-x} ]

where ( C ) is an arbitrary constant.

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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.