Thread: Integrating factor for (x^2 + y)dx - x dy = 0

1. Integrating factor for (x^3 + y)dx - x dy = 0

Can someone assist me with the integrating factor for

(x^3 + y)dx - x dy = 0 ?

I get the integrating factor (IF) as x^2

from P(x) = (∂M/∂y − N∂/∂x) / N

But Wolfram gives me the inegrating factor of 1/x.

Tammy

2. Re: Integrating factor for (x^2 + y)dx - x dy = 0

Tammyl

This is a typical linear D.E .Pls transform this to get the general form y'+p(x)y=q(x) .
if you do this correctly you will get the correct int.factor
Integrating factor - Wikipedia, the free encyclopedia

MINOAS

3. Re: Integrating factor for (x^2 + y)dx - x dy = 0

Just in case a picture helps...

First divide through by dx. The plan is to try and see the terms that involve y...

... as the result of an implicit differentiation with respect to x, via the product rule. So we want to parse them as the bottom row of this shape...

(Where the straight lines are differentiating downwards with respect to x, so that the bottom sum of two products is the derivative of the top product, as per the product rule.)

As things stand, the y terms don't quite fit the pattern...

... but the only problem with the left-forking differentiation (in the hoped-for product-rule differentiation) is the sign. The term in the lower bubble is one power lower than it's supposed anti-derivative, which gives us hope. Maybe, if we divide through by x...

Aww... but try again (divide through by x)...

Success, and we can integrate throughout with respect to x...

So much for trial and error. See here for method.

Hope that helps. Cheers!

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(x^2 y y)dx - xdy=0

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