# Did I prove this Bernoulli equation correctly?

1. Sep 16, 2006

### Pengwuino

Given a differential equation with the form:

$$\frac{{dy}}{{dx}} + P(x)y = Q(x)y^n$$

and using the substitution $$v = y^{1 - n}$$

I attempted to prove that it transforms into

$$\frac{{dv}}{{dx}} + (1 - n)P(x)v = (1 - n)Q(x)$$

Here’s the proof, did I do it correctly? I got the write answer so I assume I did :D

$$\begin{array}{l} y = v^{ - 1 + n} \\ \frac{{dv}}{{dx}} = \frac{{dv}}{{dy}}\frac{{dy}}{{dx}} \\ \frac{{dv}}{{dy}} = (1 - n)y^{ - n} \frac{{dy}}{{dx}} \\ \frac{{y^n }}{{(1 - n)}}\frac{{dv}}{{dx}} = \frac{{dy}}{{dx}} \\ \frac{{y^n }}{{(1 - n)}}\frac{{dv}}{{dx}} + P(x)y = Q(x)y^n \\ \frac{{dv}}{{dx}} + (1 - n)P(x)\frac{y}{{y^n }} = (1 - n)Q(x) \\ \frac{{dv}}{{dx}} + (1 - n)P(x)y^{1 - n} = (1 - n)Q(x) \\ \frac{{dv}}{{dx}} + (1 - n)P(x)v = (1 - n)Q(x) \\ \end{array}$$

Last edited: Sep 16, 2006