# Solving Bernoulli's Differential Equation

1. Oct 5, 2016

### Dusty912

1. The problem statement, all variables and given/known data
xy(dx)=(y2+x)dy

2. Relevant equations
integrating factor : u(x)=e∫p(x)dx
standard form of linear DE: dy/dx + P(x)y=Q(x)
standard form of bernoulli's differential equation: dy/dx + P(x)y=Q(x)yn
change of variables v=y1-n
3. The attempt at a solution
xy(dy)=(y2+x)dx
xy(dy/dx)=y2 +x
dy/dx=y/x +1/y
dy/dx-y/x=y-1

yn=y-1
y-n=y1

multipliying both sides by y1 yeilds:
y1(dy/dx) -y2/x=1

using change of variable v=y1-n=y2
dv/dx=(2y)(dy/dx)
(1/2)(dv/dx)=(y)(dy/dx)
and subbing the results yeilds:
(1/2)(dv/dx)-v/x=1

using an integrating factor to solve the resulting linear differential equation
u(x)=e∫p(x)dx
u(x)=e-∫(1/x)dx=1/x

multiplying both sides of the equation by the integrating factor yeilds:
(1/x)[(1/2)(dv/dx)-v/x]=1/x
(d/dx)[v/x]=1/x

integrating both sides:
(1/2)∫(v/x)d=∫1/xdx
(1/2)v/x=ln|x| + C
v=2ln|x| +C

now subbing back in y2]
y2]=2ln|x| + C

Unfortunately the answer is :

y2=-2x+Cx2

where did I go wrong?
Thanks ahead of time, you guys rock

2. Oct 5, 2016

### LCKurtz

Your equation is not in the proper form for the integrating factor. The leading coefficient must be $1$, so you need to multiply your equation by $2$ before continuing.

3. Oct 6, 2016

### Dusty912

thanks a bunch, I got it now