HEEELLLLLPPPP ME PLEASE
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Answer:
The level curves F(t,z) = C for any constant C in the real numbers
where
Step-by-step explanation:
Let's call
Then our differential equation can be written in the form
1) M(t,z)dt+N(t,z)dz = 0
To see that is an exact differential equation, we have to show that
2)
But
In this case we are considering t as a constant.
Similarly, now considering z as a constant, we obtain
So, equation 2) holds and then, the differential equation 1) is exact.
Now, we know that there exists a function F(t,z) such that
3)
AND
4)
We have then,
Integrating on both sides
where g(z) is a function that does not depend on t
so,
Taking the derivative of F with respect to z, we get
Using equation 4)
Hence
The function F(t,z) we were looking for is then
The level curves of this function F and not the function F itself (which is a surface in the space) represent the solutions of the equation 1) given in an implicit form.
That is to say,
The solutions of equation 1) are the curves F(t,z) = C for any constant C in the real numbers.
Attached, there are represented several solutions (for c = 1, 5 and 10)
