1. ## A differential equation

Hello

$\displaystyle xy' + y^2 = 1$, $\displaystyle y(1) = \frac{1}{3}$

My thinking:

Rearrange the equation and you will get this:

$\displaystyle \frac{1}{1 - y^{2}}\delta y = \frac{1}{x} \delta x$

Integrate it...

$\displaystyle \frac{1}{2}(\ln|1 - y| + \ln|1 + y|) = \ln|x| + C$

I think that the absolutes can be ignored, in that case I get:

$\displaystyle y = \pm \sqrt{C - x^2}$

But the correct f(x) is something very different.

2. Originally Posted by Tau

But the correct f(x) is something very different.
Probably 'cause it was set $\displaystyle C=\ln|C|$ for being constant.

3. The "answer" to this problem is:

$\displaystyle y = \frac{2x^{2} - 1}{2x^{2} + 1}$

According to the book.

Don't ask me what it means or in what context it is a "answer".

4. Originally Posted by Tau
Hello

$\displaystyle xy' + y^2 = 1$, $\displaystyle y(1) = \frac{1}{3}$

My thinking:

Rearrange the equation and you will get this:

$\displaystyle \frac{1}{1 - y^{2}}\delta y = \frac{1}{x} \delta x$

Integrate it...

$\displaystyle \frac{1}{2}({\color{red} - } \ln|1 - y| + \ln|1 + y|) = \ln|x| + C$ Mr F says: The mistake is in this line. I've added the correction (in red).

I think that the absolutes can be ignored, in that case I get:

$\displaystyle y = \pm \sqrt{C - x^2}$

But the correct f(x) is something very different.

After the correction it follows that

$\displaystyle \ln \left| \frac{1 + y}{1 - y} \right| = \ln (x^2) + 2C$

$\displaystyle \Rightarrow \frac{1 + y}{1 - y} = B x^2$

where $\displaystyle B = e^{2C}$ is also completely arbitrary.

At this stage I'd substitute $\displaystyle y(1) = \frac{1}{3}$ to get the value of B. Then I'd substitute the value for B and make y the subject.