Bernoulli Differential Equations

A first order equation $\frac{dy}{dx} = f \left( x,y \right)$ is said to be a Bernoulli equation if f(x,y) is of the form $g \left( x \right) y + h \left( x \right) y^{a}$.

Steps to solve $f \left( x,y \right) = g \left( x \right) y + h \left( x \right) y^{a}$:

1) Divide by ya to get $y^{-a} \frac{dy}{dx} = g \left( x \right) y^{1-a} + h \left( x \right)$

2) Multiply by (1 – a) to get $\left( 1-a \right) y^{-a} \frac{dy}{dx} = \left( 1-a \right) g \left( x \right) y^{1-a} + \left( 1-a \right) h \left( x \right)$

3) Let z = y(1 – a) so that equation becomes:

$$\frac{dz}{dx} = (1 – a)g(x)z + (1 – a)h(x)$$

4) Solve this linear differential equation for z.
5) Substitute back y(1 – a) = z
6) Solve the implicit solution for y.

Back To First Order Differential Equations

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