Question

the total derivative of xdy - ydx with integrating factor 1/xy is​

Answer 1

Given:

We have been given an expression, $xdy - ydx$ and Integrating Factor as $\frac{1}{xy}$.

To find:

We have to find the total derivative of the expression given above.

Solution:

According to the question, $xdy - ydx$.

We have to find the total derivative of the given expression.

It has been talked about the integrating factor, so we need to rearrange the expression in the format

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

Now, rearranging the expression,

$\Rightarrow xdy = ydx$

$\Rightarrow \frac{dy}{dx} = \frac{y}{x}$

$\Rightarrow\frac{dy}{dx} - \frac{y}{x} = 0$

Now we have to find the integrating factor.

I.F $=e^{\int \frac{1}{x}dx}=e^{log x}=x$

So, Integrating factor for the given expression is $x$.

Now,

$\Rightarrow xy=\int x\times1\ dx+C$

$\Rightarrow xy=\int x\ dx+C$

$\Rightarrow xy=\frac{x^2}{2}+C$

Therefore, total derivative is $xy=\frac{x^2}{2}+C$.