Elementary Differential Equations

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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.

We substitute $ y = v^{1/(1-4)} = v^{-1/3}$, so $ dy/dx = -(1/3)v^{-4/3}dv/dx$, and our equation becomes $$ -(1/3)v^{-4/3}\frac{dv}{dx} + 8v^{-1/3} = -6\exp(-4x)v^{-4/3} $$
Multiply by $-3v^{4/3}$ to obtain a linear equation in the usual form. $$ \frac{dv}{dx} - 24v = 18\exp(-4x) $$
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.$ $$ v = -(9/14)\exp(-4x) + C\exp(24x) $$
Back substitute for $ y.$
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.