1. Solving Bernoulli equations

I am in a class on differential equations and do not understand how to solve Bernoulli equations. Here is the problem I have been working on:

y' = ry - k(y^2) , r>0 , k>0

So far I have divided both sides by y^2, and rearranged the equation so that it looks like this:

(y')/(y^2) = (ry)/(y^2) - k(y^2)/(y^2)

(y')/(y^2) = (ry)/(y^2) - k

y'(y^-2) = r(y^-1) - k

y'(y^-2) - r(y^-1) = -k

2. Originally Posted by CatalinaLou
I am in a class on differential equations and do not understand how to solve Bernoulli equations. Here is the problem I have been working on:

y' = ry - k(y^2) , r>0 , k>0

So far I have divided both sides by y^2, and rearranged the equation so that it looks like this:

(y')/(y^2) = (ry)/(y^2) - k(y^2)/(y^2)

(y')/(y^2) = (ry)/(y^2) - k

y'(y^-2) = r(y^-1) - k

y'(y^-2) - r(y^-1) = -k

So
$y' = ry - k y^2$

then

$\frac{y' }{y^2}= \frac{r}{y} - k$

If $u = \frac{1}{y}$ then $u' = - \frac{y'}{y^2}$

$- u' = r u - k$

or

$u' - r u = k$ linear!

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