Question

xy''+y'=-4 y(1)=1 y'(1)=-8 x>0

find y' and y solutions for the IVP

Answer 1

sorry for my first comment :

y'' = (-4 -y') / x
y'' = f(x,y') -> f is independent of y
so now we can use the sub :
y' = p , y'' = dp/dx :
dp/dx = (-4-p)/x
-dp / (4+p) = dx/x

integrate
ln(4+p) = -ln(x) + C
p = A/x - 4

so y = Aln|x| -4x + B

Answer 2

now you can find A and B using the conditions.

Answer 3

Thank you!