Ordinary Differential Equations
Bernoulli Equation Abadi
Universitas Negeri Surabaya
Bernoulli equation
Bernoulli equation is one of the well known nonlinear differential equations of first order. It is written as
Where and are continuous functions.
If , the equation becomes a linear differential equation. In case of the equation becomes separable
In general case, when , bernoulli equation can be converted to a linear differential equation using the change of variable
The new differential equation for the function has the form:
Example
Find the general solution of the equation Solution:
Set , so we use the substitution
Differentiating both sides of the equation, we obtain:
Divide both sides of the original differential equation by : Subtituting and we find:
or we can write as:
We get linear equation for the function . To solve it, we use the integrating factor
The general solution of the linear equation is given by:
Returning to the function , we obtain the implicit expression:
Or
The direction fields of the that differential equation is:
exercises
Find the general solution of the equation:
1.
2. with
