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

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

asked Feb 13, 2015 in CALCULUS by anonymous

1 Answer

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Step 1 :

Definition of an Exact first order differential equation :

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

The general solution of the equation is image, 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 image.

Compare the above equation with image.

image.

Find image.

Consider image.

Apply partial derivative on each side with respect to y .

image

Consider image.

Apply partial derivative on each side with respect to x .

Since , it is an exact differential equation.

The solution of is .

Step 3 :

Find image.

Apply partial derivative on each side with respect to y .

image
From the definition .

image

Step 4 :

Now find image.

image

Substitute image in equation (1).

.

The solution is image, where C  is a constant.

The solution is image, where C  is a constant.

Solution :

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

answered Feb 25, 2015 by lilly Expert

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