I've seen plenty of proofs and exercises where people reduce a Riccati equation to a linear equation, but not the intermediate step of a Bernoulli equation. I'm trying to reduce the Riccati equation $y' = p(t) + q(t)y + r(t)y^2$ to a Bernoulli equation, which has the form $y' + p(t)y = f(t)y^n$, with the substitution $y = y_1 + u$.
I'm having some trouble understanding this particular substitution since the variables $y_1$ and $u$ are not defined. If I just follow the direction of "substitute the equation," I immediately get confused when I attempt to find $y'$ since I can't tell if we get $y' = y_1' + u'$ or $y' = 0$.
EDIT: I completely forgot to include the fact that $y_1$ is a particular solution of the equation.
Best Answer
Hint Try to think not in terms of variables, but in terms of solution functions. You had an unknown function $y(t)$ that satisfies Riccati equation. Also you have one particular solution of this equation, $y_1(t)$. Let $u(t) =y - y_1$: this is an unknown function (since $y(t)$ is unknown). What differential equation does it satisfy? Differentiate $u(t)$ and take into account that $y(t)$ and $y_1(t)$ both satisfy Riccati equation.