# Solution to Bernoulli’s Differential Equation

I cannot remember ever having to solve a Bernoulli differential equation outside of a formal course in ordinary differential equations, but I found this solution on an old thumb drive, and I happen to have written it in html format, so now you can enjoy it, too.

### Assumption

A nonlinear ODE of the form

| **(1)** |

Can be reduced to a linear differential equation using the substitution

| **(2)** |

Where n is any real number. Of course if n=1 or n=0, then the equation is already linear and no substitution is necessary.

### Proof

Start by differentiating (2) with respect to y as part of the first step in obtaining the substitution values. That is,
Solving for dy/du results in

| **(3)** |

By the chain rule the first derivative of y with respect to x can be written as

And so using (3) the first derivative of y can be rewritten as

| **(4)** |

Substituting (4) into Bernoulli’s equation produces the rather spongy looking expression

But when you multiply both sides by something interesting emerges.

| **(5)** |

Notice

Which is the same as the original substitution that

So (5) is actually just a regular old first order linear differential equation that can be solved using any known technique. In fact (5) is just the familiar standard form of a linear equation. To see this let

then (5) becomes

| **(6)** |

### Procedure for Solving a Bernoulli Differential Equation

- Massage the equation you have into the form shown in (1)
- Find P(x) and Q(x) and then substitute into (6)
- Solve the resulting first-order linear ODE
- Back substitute for u using (2) to get a solution

### Example

Solve

*Solution:*

#### Step 1 of 4

Divide both sides by x to get the form shown in (1).

#### Step 2 of 4

Note that:

Substituting P(x) and Q(x) into (6) gives

| **(7)** |

#### Step 3 of 4

The integrating factor for this last equation on, say, (0,∞) is

Multiplying both sides of (7) by this result gives

After integrating both sides you have

#### Step 4 of 4

Finally back substituting yields a solution to y