What is the difference between bernoulii equation and riccati equation
how can i decide any of them in solving 1st order non-linear ode

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 .

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