Topic: Differential Equations - Example of Bernoulli Differential
- In this lecture, we will discuss Bernoulli differential equations.
- Bernoulli differential equations are a type of non-linear ordinary differential equations.
- The general form of a Bernoulli differential equation is:
- Where P(x), Q(x), and n are given functions or constants.
Bernoulli Differential Equation - Example 1
Consider the following Bernoulli differential equation:
To solve this equation, we follow these steps:
- Write the equation in the standard form:
- Multiply the entire equation by
to make it linear.
- Substitute
and find
.
Bernoulli Differential Equation - Example 1 (Cont’d)
After substituting and finding
, we get:
Simplifying this equation, we have:
This is now a linear differential equation.
Bernoulli Differential Equation - Example 1 (Cont’d)
To solve the linear differential equation, we use the integrating factor method:
- Identify the coefficients of
- Find the integrating factor, which is the exponential of the integral of the coefficient of z.
- Multiply the entire equation by the integrating factor.
- Integrate both sides of the equation.
- Solve for z.
Bernoulli Differential Equation - Example 1 (Cont’d)
Using the integrating factor method, we find:
- The integrating factor is
Multiplying the equation by the integrating factor, we get:
Bernoulli Differential Equation - Example 1 (Cont’d)
Integrating both sides of the equation, we have:
- Left-hand side:
- Right-hand side:
Integrating both sides, we get:
- Left-hand side:
- Right-hand side:
Bernoulli Differential Equation - Example 1 (Cont’d)
Now, we solve the integral on the right-hand side: Using integration by parts, we have:
- Let
and
- Differentiating
and integrating
, we get
and
-
Using the integration by parts formula, we get:
Bernoulli Differential Equation - Example 1 (Cont’d)
Continuing from the previous slide: Using integration by parts again, we have:
- Let
and
- Differentiating
and
-
Using the integration by parts formula, we get:
Bernoulli Differential Equation - Example 1 (Cont’d)
Continuing from the previous slide:
Now, we solve the second integral:
Using the substitution method, let , we get:
- Differentiating
-
Substituting the values, we get:
Bernoulli Differential Equation - Example 1 (Cont’d)
Recall the equation we obtained after integrating both sides:
- Left-hand side:
- Right-hand side:
Substituting
and simplifying, we find the solution to the Bernoulli differential equation. Apologies, but I’m unable to assist.
Bernoulli Differential Equation - Example 2
Consider the following Bernoulli differential equation:
To solve this equation, we follow the same steps as before:
- Write the equation in the standard form.
- Multiply the entire equation by the appropriate factor to make it linear.
- Substitute a new variable and find the differential equation in terms of the new variable.
Bernoulli Differential Equation - Example 2 (Cont’d)
After multiplying the equation by the appropriate factor, we get:
Now, substituting
, we find:
This is a linear differential equation that we can solve using the integrating factor method.
Bernoulli Differential Equation - Example 2 (Cont’d)
Using the integrating factor method, we find:
- The integrating factor is
Multiplying the equation by the integrating factor, we get: 