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
$$
\begin{align}\frac{dy}{dx} - 2y &= (x + 6)y^{-3}\\
y(0) &= 4
\end{align}
$$
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-(-3)} = v^{1/4}$, so $ dy/dx = (1/4)v^{-3/4}dv/dx$, and our equation becomes $$ (1/4)v^{-3/4}\frac{dv}{dx} - 2v^{1/4} = (x + 6)v^{-3/4} $$
- Multiply by $4v^{3/4}$ to obtain a linear equation in the usual form. $$ \frac{dv}{dx} - 8v = 4x + 24 $$
- Solve the linear equation.
- Find the integrating factor $$ \mu(x) = \exp\left(\int -8 dx \right) = \exp(-8x) $$
- Multiply through by the integrating factor $$ \exp(-8x)\frac{dv}{dx} - 8\exp(-8x)v = (4x + 24)\exp(-8x) $$
- Recognize the left-hand-side as $\displaystyle \frac{d}{dx}(\mu(x)v).$ $$ \frac{d}{dx}(\exp(-8x)v) =(4x + 24)\exp(-8x) $$
- Integrate both sides. In this case you will need to integrate by parts to evaluate the integral on the right. $$ \exp(-8x)v = (-(1/2)x - 49/16)\exp(-8x) + C $$
- Divide through by $\mu(x)$ to solve for $ v.$ $$ v = -(1/2)x - 49/16+ C\exp(8x) $$

- Back substitute for $ y.$ $$ y = (-(1/2)x - 49/16+ C\exp(8x))^{1/4} $$
- 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