The Bernoulli equation is almost identical to the standard linear ODE.
\(y'=P(x)y+Q(x)y^n\)
Compare to the basic linear ODE,
\(y'=P(x)y+Q(x)\)
Meanwhile, the Riccati equation takes the form
\(y'=P(x)+Q(x)y+R(x)y^2\)
which in special cases is of Bernoulli type if \(P(x)=0\), and linear if \(R(x)=0\). But in general each type takes a different method to solve. From now on, I'll abbreviate the coefficient functions as \(P,Q,R\) for brevity.
For Bernoulli equations, the standard approach is to write
\(y'=Py+Qy^n\)
\(y^{-n}y'=Py^{1-n}+Q\)
and substitute \(v=y^{1-n}\). This makes \(v'=(1-n)y^{-n}y'\), so the ODE is rewritten as
\(\dfrac1{1-n}v'=Pv+Q\)
and the equation is now linear in \(v\).
The Riccati equation, on the other hand, requires a different substitution. Set \(v=Ry\), so that \(v'=R'y+Ry'=R'\dfrac vR+Ry'\). Then you have
\(y'=P+Qy+Ry^2\)
\(\dfrac{v'-R'\dfrac vR}R=P+Q\dfrac vR+R\dfrac{v^2}{R^2}\)
\(v'=PR+\left(Q+\dfrac{R'}R\right)v+v^2\)
Next, setting \(v=\dfrac{u'}u\), so that \(v'=\dfrac{uu''-(u')^2}{u^2}\), allows you to write this as a linear second-order equation. You have
\(\dfrac{uu''-(u')^2}{u^2}=PR+\left(Q+\dfrac{R'}R\right)\dfrac{u'}u+\dfrac{(u')^2}{u^2}\)
\(u''-\left(Q+\dfrac{R'}R\right)u'+PRu=0\)
\(u''+Su'+Tu=0\)
where \(S=-\left(Q+\dfrac{R'}R\right)\) and \(T=PR\).
