Solution of Ex:2.4 Exact differential equations by Dennis G. Zill. Exact differential equations are a type of differential equation that can be solved using a specific method known as the method of integrating factors. In the seventh edition of “Differential Equations with Boundary-Value Problems” by Dennis G. Zill, several exact differential equations are solved in exercise 2.4 on page 68. Let’s take a look at some of these solutions.

Differential Equations with BVPs

### Overview Examples of Exact DE

The first problem involves finding the solution to the differential equation . To check if this equation is exact, we can calculate the partial derivatives of with respect to x and y, and the partial derivatives of -2xy with respect to x and y. If these partial derivatives are equal, then the equation is exact. In this case, we have:

Since these partial derivatives are not equal, the equation is not exact. To make it exact, we * with suitable integrating factor.

Now,

Now, if we check the partial derivatives, we get:

Since these partial derivatives are equal, the equation is exact. We can find the potential function by integrating the coefficient of dx with respect to x and taking the derivative with respect to y, or vice versa. In this case, we get:

Taking the derivative of f with respect to y, we get:

Comparing this with the coefficient of dy, which is , we get:

Integrating both sides with respect to y, we get:

Therefore, the (GS) is:

Another problem involves finding the solution to the differential equation . In this case, the equation is already exact, since:

Therefore, we can find the potential function by integrating with respect to x and taking the derivative with respect to y, we get:

Therefore, the general solution (GS) is: