Bernoulli - Find the general solution

[EmmanuelD]

Bernoulli -- Find the general solution

Homework Statement

Find the general solution of:

y'+xy=xy^3

Homework Equations

Bernoulli's Equation

The Attempt at a Solution

y'+xy=xy^3

(y^-3)y'+x(y^-2)=x

Let v=(y^-2), thus v'=((-2y^-3)y'

Then,

-v'/2+xv=x

Multiply through by (-2)

v'-2xv=-2x

Now it's in first-order linear so I multiply by e^int(-2x)dx = e^(-x^2)

(e^(-x^2))v'-2xe^(-x^2)=2xe^(-x^2)

This is where I'm getting stuck :(

BECAUSE: the integral of e^(-x^2) is the "error function?"

According to Wolfram, integral e^(-x^2) dx = 1/2 sqrt(pi) erf(x)+constant

----

Anyone get anything different or know where I might have messed up?

THANK YOU!

 

[EmmanuelD]

Homework Statement

Find the general solution of:

y'+xy=xy^3

Homework Equations

Bernoulli's Equation

The Attempt at a Solution

y'+xy=xy^3

(y^-3)y'+x(y^-2)=x

Let v=(y^-2), thus v'=((-2y^-3)y'

Then,

-v'/2+xv=x

Multiply through by (-2)

v'-2xv=-2x

Now it's in first-order linear so I multiply by e^int(-2x)dx = e^(-x^2)

(e^(-x^2))v'-2xe^(-x^2)=2xe^(-x^2)

This is where I'm getting stuck :(

BECAUSE: the integral of e^(-x^2) is the "error function?"

According to Wolfram, integral e^(-x^2) dx = 1/2 sqrt(pi) erf(x)+constant

----

Anyone get anything different or know where I might have messed up?

THANK YOU!

GOT IT! Sorry, I don't know if there's an option for deleting a topic. My apologies.

Thanks, either way :)

 

[rude man]

Should have used separation of variables.