Question

Check whether the following differential equation is exact or not. If not, then convert it into an exact differential equation.

Answer 1

The given differential equation is not exact, if we convert it to an exact one, we get:

$2ydx + 2*(x - sin(y)^{1/2})*dy = 0$

Is the differential equation exact or not?

A differential equation:

$Ndx + Mdy = C$

Is exact only if:

$\frac{dM}{dy} = \frac{dN}{dx}$

In this case, we have:

$2ydx + (x - sin(y)^{1/2})*dy = 0\\\\then:\\\\N = 2y\\M = x - sin(y)^{1/2}$

If we differentiate, we will get:

$\frac{dN}{dy} = 2\\\\\frac{dM}{dx} = 1$

So, to convert this to an exact differential equation, we need to add a factor 2 to N, this will give:

$2ydx + 2*(x - sin(y)^{1/2})*dy = 0$

This is, in fact, an exact differential equation.

If you want to learn more about differential equations:

brainly.com/question/18760518

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