Separation of variables?

Question

Hi there,
I'd like to ask for a detailed walk through for the following problem involving the separation of variables:
dy/dx = x+3+xy+3y; y(0)=11
I'd appreciated any assistance. Thanks!

Answer 1

dy/dx = x+3+(x+3)y



dy/dx =(x+3)(1+y)



dy / (1+y) = (x+3)dx



integrate and apply x=0 y =11 to get the constant

Answer 2

dy/dx = x + 3 + xy + 3y

= 1(x + 3) + y(x + 3)

= (1 + y)(x + 3)



Separate the variables; that is, place all x's in one side and y's in the other side.

dy/dx = (y + 1)(x + 3)

dy/(y + 1) = (x + 3) dx

∫ dy/(y + 1) = ∫ (x + 3) dx

ln |y + 1| + C1 = x^2/2 + 3x + C2

x^2/2 + 3x - ln |y + 1| = C1 - C2

or x^2 + 6x - 2 ln |y + 1| = C, where C = 2(C1 - C2)



To find the value of C, use the condition y(0) = 11, or y = 11 when x = 0.

(0)^2 + 6(0) - 2 ln |11 + 1| = C

C = -2 ln 12 = ln (1/12^2) = ln (1/144)



So x^2 + 6x - 2 ln |y + 1| = ln (1/144) :)

Answer 3

y' = 1 + 0 +[1*y+x*y'] + 3y'

y' = xy' + 3y' + y + 1

hmm lol guess im not sure what ur looking for, but the derivative of that is wt i wrote above.