Solve 1st-Order DE: Riccati Equation Homework

[astrofunk21]

Homework Statement

Find the general solutions of the following 1st-order DE: y' = by²+(1/4bx²) where b ≠ 0 is a constant.

Homework Equations

Now this is a Riccati Equation, I know that. In my math class we've only learned to solve DEs the following ways:
a. Separation of Variables
b. Substitution
i. Polynomial Substitution
ii. Homogeneous Substitution
iii. Bernoulli DE

The Attempt at a Solution

The first thing I did was to assume a particular solution, which I made y₁ = A/x With this I found the derivative of it: y' = -A/x² and then y² = A²/x²

Using these I subbed them into the original equation giving me:
-A/x² = bA²/x² + 1/4bx²

After this I solved to make it equal to zero which is:
0 = A + bA² + 1/4b

From this I used the quadratic formula to get:
A = 0 and A = -1/b

After this I don't know where to go, I don't know if those are even right. Could anyone give me some direction and at the same time check if what I have so far is right?

Thanks in advance

 

[astrofunk21]

Anything?

 

[astrofunk21]

Is it that hard? Haha