Wednesday, October 21, 2015

Advanced Math Solutions – Ordinary Differential Equations Calculator, Bernoulli ODE

Last post, we learned about separable differential equations. In this post, we will learn about Bernoulli differential equation, which will require us to refresh our brains on linear first order differential equations. A Bernoulli differential equation is a differential equation that is written in the form:
y^'+p(x)y=q(x)y^n

where p(x) and q(x) are continuous functions on a given interval and n is a rational number. The concept of Bernoulli differential equations is to make a nonlinear differential equation into a linear differential equation. If n=0 or n=1, then the equation is linear. Bernoulli’s equation is used, when n is not equal to 0 or 1.


How to solve Bernoulli differential equations:
  1. Put the differential equation in the form of Bernoulli’s equation
  2. Divide by y^n
  3. Put the equation in the form \frac{1}{1-n}v^'+p(x)v=q(x)
    • We do this because we set v=y^{1-n} and v^'=(1-n)y^{-n}y^'
  4. Solve the linear first order differential equation
    • We learned how to do this in a previous post


We will solve the first example step by step (click here):


1. Put the differential equation in the form of Bernoulli’s equation


2. Divide by y^n


3. Put the equation in the form \frac{1}{1-n}v^'+p(x)v=q(x)


4. Solve linear first order differential equation




Here’s another example (click here):




As long as you memorize Bernoulli’s equation, the equation in Step 3, and how to solve linear first order differential equations, Bernoulli differential equations should be a piece of cake, even if they do take a long time to solve.

Fun fact: Jacob Bernoulli, who found Bernoulli’s equation, is brothers with Johann Bernoulli, who supposedly discovered L’Hopital’s rule. The Bernoulli family had many brilliant mathematicians.

Until next time,

Leah

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