Math 240 Home,
Textbook Contents,
Online Homework Home

**Warning: MathJax
requires JavaScript to process the mathematics on this page.**

If your browser supports JavaScript, be sure it is enabled.**
****
**

### First Order Bernoulli Equations

#### Additional Examples

Solve the following initial value problem
$$
\begin{align}\frac{dy}{dx} - 3y &= (8x + 9)y^{-2}\\
y(0) &= 1
\end{align}
$$
This is a Bernoulli equation. First we find the general solution following the paradigm.
*You may reload this page to generate additional examples.*

If you have any problems with this page, please contact bennett@math.ksu.edu.

©2010, 2014 Andrew G. Bennett

If your browser supports JavaScript, be sure it is enabled.

- We substitute $ y = v^{1/(1-(-2)} = v^{1/3}$, so $ dy/dx = (1/3)v^{-2/3}dv/dx$, and our equation becomes $$ (1/3)v^{-2/3}\frac{dv}{dx} - 3v^{1/3} = (8x + 9)v^{-2/3} $$
- Multiply by $3v^{2/3}$ to obtain a linear equation in the usual form. $$ \frac{dv}{dx} - 9v = 24x + 27 $$
- Solve the linear equation.
- Find the integrating factor $$ \mu(x) = \exp\left(\int -9 dx \right) = \exp(-9x) $$
- Multiply through by the integrating factor $$ \exp(-9x)\frac{dv}{dx} - 9\exp(-9x)v = (24x + 27)\exp(-9x) $$
- Recognize the left-hand-side as $\displaystyle \frac{d}{dx}(\mu(x)v).$ $$ \frac{d}{dx}(\exp(-9x)v) =(24x + 27)\exp(-9x) $$
- Integrate both sides. In this case you will need to integrate by parts to evaluate the integral on the right. $$ \exp(-9x)v = (-(8/3)x - 89/27)\exp(-9x) + C $$
- Divide through by $\mu(x)$ to solve for $ v.$ $$ v = -(8/3)x - 89/27+ C\exp(9x) $$

- Back substitute for $ y.$ $$ y = (-(8/3)x - 89/27+ C\exp(9x))^{1/3} $$
- We check, but $ y = 0$ is not a solution to this equation.

If you have any problems with this page, please contact bennett@math.ksu.edu.

©2010, 2014 Andrew G. Bennett