Bernoulli Differential Equations

A first order equation $\frac{dy}{dx} = f \left( x,y \right)$ is said to be a Bernoulli equation if f(x,y) is of the form $g \left( x \right) y + h \left( x \right) y^{a}$.

Steps to solve $f \left( x,y \right) = g \left( x \right) y + h \left( x \right) y^{a}$:

1) Divide by ya to get $y^{-a} \frac{dy}{dx} = g \left( x \right) y^{1-a} + h \left( x \right)$

2) Multiply by (1 – a) to get $\left( 1-a \right) y^{-a} \frac{dy}{dx} = \left( 1-a \right) g \left( x \right) y^{1-a} + \left( 1-a \right) h \left( x \right)$

3) Let z = y(1 – a) so that equation becomes:


$$ \frac{dz}{dx} = (1 – a)g(x)z + (1 – a)h(x)$$

4) Solve this linear differential equation for z.
5) Substitute back y(1 – a) = z
6) Solve the implicit solution for y.

Back To First Order Differential Equations

Back To University Year 1 Physics Notes

Mini Physics

Administrator of Mini Physics. If you spot any errors or want to suggest improvements, please contact us. If you like the content in this site, please recommend this site to your friends!

Leave a Comment

This site uses Akismet to reduce spam. Learn how your comment data is processed.