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Exact differential equations

Screen Shot 2016-04-03 at 13.47.46.png

Could someone tell me simply, what is an exact differential equation.
For this example, how do i determine whether it is an exact differential equation or not.

Does it have to be in this form to be exact.
Screen Shot 2016-04-03 at 13.51.54.png
(edited 8 years ago)
Reply 1
Avatar for B_9710
B_9710
18
Is this the same as the differential equation above that you posted ?
ddx[x3y]=x \displaystyle \frac{d}{dx} \left [x^3y \right ] = x
The LHS is exactly the derivative of x3y x^3y .
(edited 8 years ago)
Reply 2
Avatar for Zacken
Zacken
22
Original post by Fenugreek

Does it have to be in this form to be exact.
Screen Shot 2016-04-03 at 13.51.54.png


Yes, an exact differential equation is where one side can be re-written as ddx(yf(x))\displaystyle \frac{\mathrm{d}}{\mathrm{d}x } \left( yf(x) \right)

In this case, your DE is exact, because x3dydx+3x2y=ddx(x3y)\displaystyle x^3 \frac{\mathrm{d}y}{\mathrm{d}x} + 3x^2 y = \frac{\mathrm{d}}{\mathrm{d}x } \left( x^3y\right).

You can check this equality is true by differentiating x3yx^3y with respect to xx using the product rule.

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