Question

Verify that the differential equation:
(3y cos(xy) - 2xy² - e^-x) dx + (3x cos(xy) - 2x²y - e^-y) dy = 0;
is exact and then solve it.

Answer 1

Verify the given differential equation is exact and solve.

Given  

The given differential equation is of the form:

Where,

and

Any given differential equation of the form as in equation (1) is said to be exact if:

Therefore, finding partial differential of first, we get:

or,

Similarly, calculating partial differential of , we get:

Since, the values of and are equal, therefore we can conclude that the given differential equation is exact.

The solution for an exact differential equation is given by:

Note that the solution of is written without the terms containing .

Integrating first, we get:

Integrating the above, we get:

Integrating , we get:

Integrating the above, we get:

Eliminating all the terms containing , we are left with the following:

Using equations (2), (3) and (4), the solution to the given differential equation is as follows: