Given a differential equation with the form:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{{dy}}{{dx}} + P(x)y = Q(x)y^n [/tex]

and using the substitution [tex]v = y^{1 - n}[/tex]

I attempted to prove that it transforms into

[tex]\frac{{dv}}{{dx}} + (1 - n)P(x)v = (1 - n)Q(x)[/tex]

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

[tex]\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}

[/tex]

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Did I prove this Bernoulli equation correctly?

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**