   # Integrating FactorShow that thedifferential equation ( a x y 2 + b y ) d x + ( a x 2 y + a x ) d y = 0 is exact only when a = b . For a ≠ b , show that x m y n is an integrating factor, where m = − 2 b + a a + b , n = − 2 a + b a + b ### Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378

#### Solutions

Chapter
Section ### Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378
Chapter 16.1, Problem 38E
Textbook Problem
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## Integrating FactorShow that thedifferential equation ( a x y 2 + b y ) d x + ( a x 2 y + a x ) d y = 0 is exact only when a = b . For a ≠ b , show that x m y n is an integrating factor, where m = − 2 b + a a + b , n = − 2 a + b a + b

To determine

To prove: The differential equation (axy2+by)dx+(bx2y+ax)dy=0 is exact only when a=b and also show that if ab then xmyn is an integrating factor where m=2b+aa+b,n=2a+ba+b.

### Explanation of Solution

Concept Used:

Let M and N have continuous partial derivatives on an open disk R. The differential equation

M(x,y)dx+N(x,y)dy=0

is exact if and only if

My=Nx

Proof:

For the given differential equation (axy2+by)dx+(bx2y+ax)dy=0

M=(axy2+by)N=(bx2y+ax)My=(axy2+by)yMy=2axy+bNx=(bx2y+ax)xNx=2bxy+a

It is given that the above equation is exact, therefore My=Nx

After equation the values we get:

2axy+b=2bxy+a2axy2bxya+b=02xy(ab)(ab)=0(ab)(2xy1)=0

We got two solutions a=b and xy=12. Since xy=12 is a particular solution of the general differential equation, which cannot be considered as solution, therefore we must consider a=b as solution.

To prove that if ab then xmyn is an integrating factor, we have to multiply the differential equation (axy2+by)dx+(bx2y+ax)dy=0 by xmyn.

(axy2+by)(xmyn)dx+(bx2y+ax)(xmyn)dy=0

Now we can compare the differential equation (axy2+by)(xmyn)dx+(bx2y+ax)(xmyn)dy=0 with the M(x,y)dx+N(x,y)dy=0

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