solve the following exact ordinary differential equation:
(2tz^3 + ze^(tz) - 4) dt + (3t^2z^2 + te^(tz) + 2) dz = 0
^3 = to the power of 3

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)

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