Bernoulli Differential equation?

Question

I need to solve these 2 questions:

1. Find the solution to the following diff. equation
(x+y)dy/dx + y = 0
the answer should be y^2+2xy=C

2. Wich is the solution to the following differential equation?
dy/dx - (1/x)y = -xy^2
the answer should be x/(2x^2 + C)

I tried working them according to what the book shows me but get stuck. I used Q(x)= -x P(x)= -1/x
let n = 2

Who can help?

Answer 1

1. Solving for dy/dx gives dy/dx = -y/(x+y), but this doesn't fit the form of the Bernoulli equation. Instead, flip it around and get dx/dy + (1/y)x = -1, so n=0, P(y)=1/y, and Q(y)=-1. In this case, there's no need for a change of variable, so instead just jump directly to using the integrating factor m(y) = e^(integral of 1/y) = y, to get

y dx/dy + x = -y, so (xy)' = -y. Therefore

x = (1/y) (-y^2/2 + C), or x = -y/2 + C/y.

Multiply through by 2y: 2xy + y^2 = 2C, which is equivalent to the solution you expect, since C is an arbitrary constant.

2. Since n=2, let v = y^(1-n) = 1/y. After transforming, you get

dv/dx + vp(x) = q(x)

where p(x) = (1-n)P(x) = 1/x and q(x) = (1-n)Q(x) = x, so you're solving

dv/dx + (1/x)v = x.

Now use the integrating factor x, and you have

x dv/dx + v = x^2

so (vx)' = x^2, or v = (1/x)(x^3/3 + C), so

v = x^2 / 3 + C/x. But v = 1/y, so

y = 1 / (x^2/3 + C/x) = x / (x^3/3 + C).

This doesn't agree with what you're saying the answer should be, but it does satisfy the differential equation (as I confirmed by plugging in this solution), whereas your answer doesn't.

Answer 2

1. Solving for dy/dx gives dy/dx = -y/(x+y).
flip it around and get dx/dy + (1/y)x = -1, so n=0, P(y)=1/y, and Q(y)=-1.
then m(y) = e^(integral of 1/y) = y,
so
y dx/dy + x = -y, and so (xy)' = -y.
Therefore
x = (1/y) (-y^2/2 + C), or x = -y/2 + C/y.

2. n=2, let v = y^(1-n) = 1/y. Then
dv/dx + vp(x) = q(x)
where p(x) = (1-n)P(x) = 1/x and q(x) = (1-n)Q(x) = x, so
dv/dx + (1/x)v = x.
=>
x dv/dx + v = x^2
=>
(vx)' = x^2, or v = (1/x)(x^3/3 + C),
=>
v = x^2 / 3 + C/x.
But v = 1/y, =>
y = 1 / (x^2/3 + C/x) = x / (x^3/3 + C).

s