# Bernoulli Equation

Bernoulli equation is one of the well known nonlinear differential equations of the first order. It is written as

where *a* (*x*) and *b* (*x*) are continuous functions.

If \(m = 0,\) the equation becomes a linear differential equation. In case of \(m = 1,\) the equation becomes separable.

In general case, when \(m \ne 0,1,\) Bernoulli equation can be converted to a linear differential equation using the change of variable

The new differential equation for the function \(z\left( x \right)\) has the form:

and can be solved by the methods described on the page Linear Differential Equation of First Order.

## Solved Problems

### Example 1.

Find the general solution of the equation \[y' - y = {y^2}{e^x}.\]

Solution.

We set \(m = 2\) for the given Bernoulli equation, so we use the substitution

Differentiating both sides of the equation (we consider \(y\) in the right side as a composite function of \(x\)), we obtain:

Divide both sides of the original differential equation by \({y^2}:\)

Substituting \(z\) and \(z',\) we find

We get the linear equation for the function \(z\left( x \right).\) To solve it, we use the integrating factor:

Then the general solution of the linear equation is given by

Since \(C\) is an arbitrary constant, we can replace \(2C\) with a constant \(C_1.\) Returning to the function \(y\left( x \right),\) we obtain the implicit expression:

Note that we have lost the solution \(y = 0\) when dividing the equation by \({y^2}.\) Thus, the final answer is given by

### Example 2.

Solve the differential equation \[y' + \frac{y}{x} = {y^2}.\]

Solution.

As it can be seen, this differential equation is a Bernoulli equation. To solve it, we make the substitution

Differentiating, we find:

Divide the original equation by \({y^2}\) and replace \(y\) with \(z:\)

When dividing by \({y^2},\) we have lost the solution \(y = 0.\) (You can check this by direct substitution.)

In terms of \(z,\) the differential equation is written in the form:

We get the linear equation for the function \(z\left( x \right),\) so we can solve it using the integrating factor technique:

We can make sure that the function \(\frac{1}{x}\) is the integrating factor. Indeed:

We see that the left side of the equation becomes the derivative of the product \(z\left( x \right)u\left( x \right)\) after multiplying by \(\frac{1}{x}\).

Then the general solution of the linear equation for \(z\left( x \right)\) is given by

Taking into account that \(y = \frac{1}{z},\) we can write the answer:

or in the implicit form:

Thus, the final answer is