1. Bernoulli Differential Equations
AP Calculus BC

2. Bernoulli Differential Equations
A Bernoulli differential equation can is of the form where P and Q are continuous functions on a given interval.
Note that if n = 0 or 1, then we have a linear differential equation.
But what if n > 1?
Our goal is to take this non-linear differential equation and turn it into a linear differential equation, so we can solve it.

3. Reduction Divide by yn 
We’re going to use u-substitution, so let u = y1–n  So
Substitute back into original equation 
Multiply by (1 – n) 
The differential equation is now linear (ugly, but linear)!

4. Example 1 Find the general solution to the differential equation
1) n = 2, so u = y1–2, or u = y–1.
2) Divide equation by y2 to get RHS to equal x 
3) Find du/dx 
4) Solve for dy/dx 

5. Example 1 (cont.)
5) Substitute into equation in #2  6) Now this is a linear differential equation, so solve using integrating factor.

6. Example 1 (cont.)
7) Multiply by x–1  8) Product Rule in reverse  9) Integrate  10) Multiply by x  11) Substitute back for y (Recall u = y–1) y–1 = –x2 + Cx 

7. Example 2
Find the general solution to

8. Example 3 – Initial Value Problem
Solve the differential equation with the initial condition y(1) = 0.

9. Still Confused? Watch this video:
It helped me!