Differential Equations help (1 Viewer)

notme123 · Sep 25, 2021

Please?

Grammar 2021 btw

Drongoski · Sep 25, 2021

$\frac{dy}{dx} =\frac {d}{dx}xu = u + x\frac{du}{dx}\\ \\ y = xu \implies u = \frac {y}{x} \\ \\ \frac {dy}{dx} = \frac{1}{2}((\frac{y}{x})^2 + 1) \\ \\ \therefore u + x\frac{du}{dx} = \frac{1}{2}(u^2 + 1) \\ \\ \therefore x\frac{du}{dx} = \frac{1}{2}(u^2 -2u + 1) = \frac{1}{2}(u-1)^2\\ \\ \therefore \frac{1}{x}dx = \frac{2}{(u-1)^2 }du \\ \\ \therefore \int \frac{1}{x} dx = \int \frac{2}{(u-1)^2}du$

etc etc

$\therefore ln|x| = -\frac{2}{u-1} + C = -(\frac{2}{\frac{y}{x} - 1}) + C\\ \\ y(1) = 2 \implies ln|1| = -\frac {2}{\frac{2}{1} -1} + C = 2 \implies C = 2$

With some fiddling, you get (I think):

$y = x + \frac{2x}{2 - ln|x|}$

CM_Tutor · Sep 25, 2021

Drongoski said:
$\frac{dy}{dx} =\frac {d}{dx}xu = u + \frac{du}{dx}$

This should be using the Product Rule, so that

$\frac{dy}{dx} =\frac{d}{dx}\big(xu\big) = u \times 1 + x \times \frac{du}{dx} = u + x\cfrac{du}{dx}$

which I see is used correctly later on, so this is just a typo.

Drongoski · Sep 25, 2021

CM_Tutor said:
This should be using the Product Rule, so that

$\frac{dy}{dx} =\frac{d}{dx}\big(xu\big) = u \times 1 + x \times \frac{du}{dx} = u + x\cfrac{du}{dx}$

which I see is used correctly later on, so this is just a typo.

Thanks CM_Tutor for pointing out my typo.

Some of you may have noticed I struggle one line at a time when using LaTeX, for a reason.

notme123 · Sep 25, 2021

Drongoski said:
$\frac{dy}{dx} =\frac {d}{dx}xu = u + x\frac{du}{dx}\\ \\ y = xu \implies u = \frac {y}{x} \\ \\ \frac {dy}{dx} = \frac{1}{2}((\frac{y}{x})^2 + 1) \\ \\ \therefore u + x\frac{du}{dx} = \frac{1}{2}(u^2 + 1) \\ \\ \therefore x\frac{du}{dx} = \frac{1}{2}(u^2 -2u + 1) = \frac{1}{2}(u-1)^2\\ \\ \therefore \frac{1}{x}dx = \frac{2}{(u-1)^2 }du \\ \\ \therefore \int \frac{1}{x} dx = \int \frac{2}{(u-1)^2}du$

etc etc

damn, I was closish. I did the u=y/x thing but only on one of the powers of y/x for some stupid reason.

Differential Equations help (1 Viewer)

notme123

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Drongoski

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CM_Tutor

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Drongoski

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notme123

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