# Exact Differential Equation

• Feb 4th 2009, 02:30 PM
mandy123
Exact Differential Equation
I am told to verify that the given differential equation is exact; then solve it:
(x + tan-1y)dx + ((x + y)/(1 + y^2))dy=0

So far I have found the exactness and have gotten down to
F= (x^2) + 2xtan^(-1) (y) + g(y)

I am then trying to find g(y) and to do this do I have to set
F= (x^2) + 2xtan^(-1) (y) + g(y) = ((x + y)/(1 + y^2))
If I do, how in the world do I cancel things out to get the anwer!

The answer is suppose to be
(x^2) + 2xtan^(-1) (y) + ln(1+y^2)

Thanks
• Feb 4th 2009, 04:00 PM
Jester
Quote:

Originally Posted by mandy123
I am told to verify that the given differential equation is exact; then solve it:
(x + tan-1y)dx + ((x + y)/(1 + y^2))dy=0

So far I have found the exactness and have gotten down to
F= (x^2) + 2xtan^(-1) (y) + g(y)

I am then trying to find g(y) and to do this do I have to set
F= (x^2) + 2xtan^(-1) (y) + g(y) = ((x + y)/(1 + y^2))
If I do, how in the world do I cancel things out to get the anwer!

The answer is suppose to be
(x^2) + 2xtan^(-1) (y) + ln(1+y^2)

Thanks

If $M dx + N dy = 0$ is exact then $M_y = N_x$ (which you did). If so, then F exist such that

$F_x = m,\;\;\;F_y = N$

$F_x = x + \tan^{-1}y,\;\;\;F_y = \frac{x+y}{1+y^2}$
$F = \frac{x^2}{2} + x\tan^{-1}y + g(y)$
$F_y = \frac{x+y}{1+y^2}$