First Order Bernoulli Equations
Additional Examples
- We substitute $ y = v^{1/(1-3)} = v^{-1/2}$, so $ dy/dx = -(1/2)v^{-3/2}dv/dx$, and our equation becomes $$ -(1/2)v^{-3/2}\frac{dv}{dx} + 5v^{-1/2} = -10\exp(-5x)v^{-3/2} $$
- Multiply by $-2v^{3/2}$ to obtain a linear equation in the usual form. $$ \frac{dv}{dx} - 10v = 20\exp(-5x) $$
- Solve the linear equation.
- Find the integrating factor $$ \mu(x) = \exp\left(\int -10 dx \right) = \exp(-10x) $$
- Multiply through by the integrating factor $$ \exp(-10x)\frac{dv}{dx} - 10\exp(-10x)v = 20\exp(-5x)\exp(-10x) = 20\exp(-15x) $$
- Recognize the left-hand-side as $\displaystyle \frac{d}{dx}(\mu(x)v).$ $$ \frac{d}{dx}(\exp(-10x)v) =20exp(-15x) $$
- Integrate both sides. $$ \exp(-10x)v = -(4/3)\exp(-15x) + C $$
- Divide through by $\mu(x)$ to solve for $ v.$ $$ v = -(4/3)\exp(-5x) + C\exp(10x) $$
- Back substitute for $ y.$ $$ y = (-(4/3)\exp(-5x) + C\exp(10x))^{-1/2} $$
- We check that $ y = 0$ is indeed a singular solution.
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©2010, 2014 Andrew G. Bennett