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Marizza181 [45]
2 years ago
14

Determine whether the given differential equation is exact. If it is exact, solve it. (tan(x)-sin(x)sin*y))dx+cos(x)cos(y)dy=0 g

Mathematics
1 answer:
Nataly [62]2 years ago
7 0

The differential equation

M(x,y) \, dx + N(x,y) \, dy = 0

is considered exact if M_y = N_x (where subscripts denote partial derivatives). If it is exact, then its general solution is an implicit function f(x,y)=C such that f_x=M and f_y=N.

We have

M = \tan(x) - \sin(x) \sin(y) \implies M_y = -\sin(x) \cos(y)

N = \cos(x) \cos(y) \implies N_x = -\sin(x) \cos(y)

and M_y=N_x, so the equation is indeed exact.

Now, the solution f satisfies

f_x = \tan(x) - \sin(x) \sin(y)

Integrating with respect to x, we get

\displaystyle \int f_x \, dx = \int (\tan(x) - \sin(x) \sin(y)) \, dx

\implies f(x,y) = -\ln|\cos(x)| + \cos(x) \sin(y) + g(y)

and differentiating with respect to y, we get

f_y = \cos(x) \cos(y) = \cos(x) \cos(y) + \dfrac{dg}{dy}

\implies \dfrac{dg}{dy} = 0 \implies g(y) = C

Then the general solution to the exact equation is

f(x,y) = \boxed{-\ln|\cos(x)| + \cos(x) \sin(y) = C}

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at the mall, myla bought a shirt for $25.36, a hat for $5.09 and a belt for $11.45. how much did she spend altogether?
malfutka [58]

The summation of the money Myla spend altogether is $41.9

As in the given problem  we can see Myla bought

a shirt for $25.36 and

a hat for  $5.09  and

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Hence we need to know the cummulative price she spend altogether.

To get the total price we should add the price of shirt hat and belt.

Summing all the prices we get

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