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Math 240 Home, Textbook Contents, Online Homework Home

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### First Order Bernoulli Equations

#### Additional Examples

Solve the following initial value problem $$\begin{align}\frac{dy}{dx} + 8y &= -6\exp(-4x)y^{4}\\ y(0) &= 3 \end{align}$$ This is a Bernoulli equation. First we find the general solution following the paradigm.

1. We substitute $y = v^{1/(1-4)} = v^{-1/3}$, so $dy/dx = -(1/3)v^{-4/3}dv/dx$, and our equation becomes $$-(1/3)v^{-4/3}\frac{dv}{dx} + 8v^{-1/3} = -6\exp(-4x)v^{-4/3}$$
2. Multiply by $-3v^{4/3}$ to obtain a linear equation in the usual form. $$\frac{dv}{dx} - 24v = 18\exp(-4x)$$
3. Solve the linear equation.

1. Find the integrating factor $$\mu(x) = \exp\left(\int -24 dx \right) = \exp(-24x)$$
2. Multiply through by the integrating factor $$\exp(-24x)\frac{dv}{dx} - 24\exp(-24x)v = 18\exp(-4x)\exp(-24x) = 18\exp(-28x)$$
3. Recognize the left-hand-side as $\displaystyle \frac{d}{dx}(\mu(x)v).$ $$\frac{d}{dx}(\exp(-24x)v) =18exp(-28x)$$
4. Integrate both sides. $$\exp(-24x)v = -(9/14)\exp(-28x) + C$$
5. Divide through by $\mu(x)$ to solve for $v.$ $$v = -(9/14)\exp(-4x) + C\exp(24x)$$

4. Back substitute for $y.$
5. $$y = (-(9/14)\exp(-4x) + C\exp(24x))^{-1/3}$$
6. We check that $y = 0$ is indeed a singular solution.

Now we plug in the initial values $x = 0$ and $y = 3$ and solve for $C = 257/378,$ to obtain the solution to the initial value problem $$y = ( -(9/14)\exp(-4x) + (257/378)\exp(24x) )^{-1/3}$$ You may reload this page to generate additional examples.
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