I really need help with math homework. Please and thank you!

Question

Find the domain for the particular solution to the differential equation , with initial condition y(2) = 2.

Options:

1)	x > -2
2)
3)	[-4, 4]
4)	All real numbers

Answer 1

dy/dx = -xy

dy/y = -x dx

ln y = (-1/2)x^2 + c

y = exp [( -1/2) x^2 + c ]

y(2)=2 ---> (2,2)

2 = exp[ (-1/2) 2^2 + c]

2 = exp ( c-2)

ln 2 = c-2

c = ln2 + 2

y = exp [ (-1/2) x^2 + ln 2 + 2]

check by diff:

y' = exp [ (-1/2)x^2 + ln2 + 2] [ -x]

= -xy