#2. Use substitution to solve the system of equations x=3y 34-5y =12 Solution: (
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The Bernoulli equation is almost identical to the standard linear ODE.
Compare to the basic linear ODE,
Meanwhile, the Riccati equation takes the form
which in special cases is of Bernoulli type if
, and linear if 
. But in general each type takes a different method to solve. From now on, I'll abbreviate the coefficient functions as 
for brevity.
For Bernoulli equations, the standard approach is to write
and substitute
. This makes 
, so the ODE is rewritten as
and the equation is now linear in
.
The Riccati equation, on the other hand, requires a different substitution. Set
, so that 
. Then you have
Next, setting
, so that 
, allows you to write this as a linear second-order equation. You have
where
and 
.
Compare to the basic linear ODE,
Meanwhile, the Riccati equation takes the form
which in special cases is of Bernoulli type if
For Bernoulli equations, the standard approach is to write
and substitute
and the equation is now linear in
The Riccati equation, on the other hand, requires a different substitution. Set
Next, setting
where