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### First Order Bernoulli Equations

#### Additional Examples

Solve the following initial value problem
$$
\begin{align}\frac{dy}{dx} + y &= -7\exp(-8x)y^{4}\\
y(0) &= 4
\end{align}
$$
This is a Bernoulli equation. First we find the general solution following the paradigm.
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©2010 Andrew G. Bennett

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- We substitute $ y = v^{1/(1-4)} = v^{-1/3}$, so $ dy/dx = -(1/3)v^{-4/3}dv/dx$, and our equation becomes $$ -(1/3)v^{-4/3}\frac{dv}{dx} + v^{-1/3} = -7\exp(-8x)v^{-4/3} $$
- Multiply by $-3v^{4/3}$ to obtain a linear equation in the usual form. $$ \frac{dv}{dx} - 3v = 21\exp(-8x) $$
- Solve the linear equation.
- Find the integrating factor $$ \mu(x) = \exp\left(\int -3 dx \right) = \exp(-3x) $$
- Multiply through by the integrating factor $$ \exp(-3x)\frac{dv}{dx} - 3\exp(-3x)v = 21\exp(-8x)\exp(-3x) = 21\exp(-11x) $$
- Recognize the left-hand-side as $\displaystyle \frac{d}{dx}(\mu(x)v).$ $$ \frac{d}{dx}(\exp(-3x)v) =21exp(-11x) $$
- Integrate both sides. $$ \exp(-3x)v = -(21/11)\exp(-11x) + C $$
- Divide through by $\mu(x)$ to solve for $ v.$ $$ v = -(21/11)\exp(-8x) + C\exp(3x) $$

- Back substitute for $ y.$ $$ y = (-(21/11)\exp(-8x) + C\exp(3x))^{-1/3} $$
- We check that $ y = 0$ is indeed a singular solution.

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

©2010 Andrew G. Bennett