(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The question is to solve the inexact equation by turning it into exact.the equation is ##( x + y + 4 ) d x + ( - x + y + 6 ) d y = 0##

Where "x" and "y" are variable.

2. Relevant equations

1.(x+y+4)=m and (-x+y+6)=n

2.Integrating Factor =##\frac { 1 } { x ^ { 2 } + y ^ { 2 } + 4 x + 6 y }##

3.a=[(Integrating factor)*m]=$$\frac { ( x + y + 4 ) } { x ^ { 2 } + y ^ { 2 } + 4 x + 6 y }$$ and b=[(Integrating factor)*m]=$$\frac { ( x + y + 4 ) } { x ^ { 2 } + y ^ { 2 } + 4 x + 6 y }$$

4.##t = (x ^ { 2 } + y ^ { 2 } + 4 x + 6 y)##

3. The attempt at a solution

The given equation is a non exact homogeneous equation and is in the form

mdx+ndy = 0

now let's take

( x + y + 4 ) = m and ( - x + y + 6 ) = n

Here ##m x+n y##

=## x ^ { 2 } + x y + 4 x - x y + y ^ { 2 } + 6 y##

= ##x ^ { 2 } + y ^ { 2 } + 4 x + 6 y \neq 0##

So the integrating factor of the equation should be

I.F = ##\frac { 1 } { m x + n y } = \frac { 1 } { x ^ { 2 } + y ^ { 2 } + 4 x + 6 y }##

So now let's convert the inexact equation to exact by multiplying each term with integrating factor.so the new exact equation should be

##\frac { ( x + y + 4 ) } { x ^ { 2 } + y ^ { 2 } + 4 x + 6 y } + \frac { ( - x + y + 6 ) } { x ^ { 2 } + y ^ { 2 } + 4 x + 6 y } = 0##

=##a x + b y =0##

Where a=$$\frac { ( x + y + 4 ) } { x ^ { 2 } + y ^ { 2 } + 4 x + 6 y }$$ and b= $$\frac { ( x + y + 4 ) } { x ^ { 2 } + y ^ { 2 } + 4 x + 6 y }$$

Now the solution should be ##\int a d x + \int b d y = constant##

Where [in term "a" y is constant and the term "b" is free from x]

=##\int \frac { ( x + y + 4 ) d x } { x ^ { 2 } + y ^ { 2 } + 4 x + 6 y } = C##

(As there is no term in "b" free from x)

(##C##=constant)

So$$ \int \frac { ( x + y + 4 ) d x } { x ^ { 2 } + y ^ { 2 } + 4 x + 6 y } = $$

=$$\int \frac { x + 2 } { x ^ { 2 } + y ^ { 2 } + 4 x + 6 y } + \int \frac { ( y + 2 ) d x } { x ^ { 2 } + y ^ { 2 } + 4 x + 6 y }$$

[Now let's put ##(x ^ { 2 } + y ^ { 2 } + 4 x + 6 y) = t##

So ##( x + 2 ) d x = \frac { d t } { 2 }##]

=$$\frac { 1 } { 2 } \int \frac { d t } { t } + (\int \frac { ( y + 2 ) d x } { x ^ { 2 } + y ^ { 2 } + 4 x + 6 y } )$$

=$$\frac { 1 } { 2 } \log t + ( \int \frac { ( y + 2 ) d x } { x ^ { 2 } + y ^ { 2 } + 4 x + 6 y }) $$

=$$\log t ^ { 2 } + ( \int \frac { ( y + 2 ) d x } { x ^ { 2 } + y ^ { 2 } + 4 x + 6 y })$$

=$$\log ( x ^ { 2 } + ( y ^ { 2 } + 4 x + 6 y ) ^ { 2 } + (\int \frac { ( y + 2 ) d x } { x ^ { 2 } + y ^ { 2 } + 4 x + 6 y }) $$

So now ##\log ( x ^ { 2 } + y ^ { 2 } + 4 x + 6 y ) ^ { 2 } + ( \int \frac { ( y + 2 ) d x } { x ^ { 2 } + y ^ { 2 } + 4 x + 6 y }) ##= constant

Now I can't figure out what to do next.please help.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Help in solving an inexact differential equation

Have something to add?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**