Question

Solve the exact differential equation: [ye^x + cos(x)] dx + [e^x + (1/y)] dy = 0 Thank you!?

Answer 1

Step 1 :

Definition of an Exact first order differential equation :

The equation is an exact differential equation if there exists a function f  of two variables x  and y  having continuous partial derivatives such that .

The general solution of the equation is , where C  is a constant and .

The function should be there, since in integration, assume that the variable y  is constant.

Step 2 :

The eaxct differential equation is .

Compare the above equation with .

.

Find .

Consider .

Apply partial derivative on each side with respect to y .

Consider .

Apply partial derivative on each side with respect to x .

Since , it is an exact differential equation.

The solution of is .

Step 3 :

Find .

Apply partial derivative on each side with respect to y .

From the definition .

Step 4 :

Now find .

Substitute in equation (1).

.

The solution is , where C  is a constant.

The solution is , where C  is a constant.

Solution :

The solution of the differential equation is , where C  is a constant.