# Different answer for diffrential equations using separable and bernoulli's equation

Okay so I have a simple diffrential equation where I get two different answers depending on the method I use. Additionally, Wolfram has yet a other solution. The differential equation to be solved is:

$y'+ ry =$$-\frac{ry^2}{b}$

So using the Bernoulli's equation we have the following substitution:

$y = z^{-1}$ -> $z'-rz = \frac{r}{b}$

Solving the equation gives:

$\frac{1}{e^{rt}-\frac{1}{b}}$

On the other hand if separated equation is used, the equation becomes:

$\frac{y'}{-ry-\frac{ry^2}{b}} = 1$

Integrating we have:

$\frac{ln\frac{y+b}{y}}{r}=t$

Solving the equation:

$y=\frac{-b}{1-e^{rt}}$

Any ideas where I have gone wrong? Wolfram alpha gives the following answer:

$-\frac{be^b}{e^b-e^{rt}}$

• What phappened to $a$? It does not appear in the solution. – Julián Aguirre Dec 11 '15 at 10:37
• It looks like the $a$ should be $r$ and there is an initial condition you didn't write. – KittyL Dec 11 '15 at 10:40
• @JuliánAguirre Apologies it's r and not a. Good catch. – Dole Dec 11 '15 at 10:40
• What initial conditions are you using in the different solutions? – Mankind Dec 11 '15 at 11:15
• And you have also forgotten the constant of integration in both your solutions (which I believe is the root of all your troubles). – Hans Lundmark Dec 11 '15 at 11:31