Question

xdx+ydy=a^2(xdy-ydx)/(x^2+y^2)

Answer 1

Simplify —— x2

Dividing exponential expressions :

 1.1    x1 divided by x2 = x(1 - 2) = x(-1) = 1/x1 = 1/x

Equation at the end of step  1  :

1 (x2d + dy2) - ((xdy - (dy • —)) + y2) = 0 x

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole 

Rewrite the whole as a fraction using  x  as the denominator :

xdy xdy • x xdy = ——— = ——————— 1 x

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole 

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 2.2       Adding up the two equivalent fractions 
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

xdy • x - (dy) x2dy - dy —————————————— = ————————— x x

Equation at the end of step  2  :

(x2dy - dy) (x2d + dy2) - (——————————— + y2) = 0 x

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a whole to a fraction 

Rewrite the whole as a fraction using  x  as the denominator :

y2 y2 • x y2 = —— = —————— 1 x

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   x2dy - dy  =   dy • (x2 - 1) 

Trying to factor as a Difference of Squares :

 4.2      Factoring:  x2 - 1 

Theory : A difference of two perfect squares,  A2 - B2  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A2 - AB + BA - B2 =
         A2 - AB + AB - B2 = 
         A2 - B2

Note :  AB = BA is the commutative property of multiplication. 

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 1 is the square of 1
Check :  x2  is the square of  x1 

Factorization is :       (x + 1)  •  (x - 1) 

Adding fractions that have a common denominator :

 4.3       Adding up the two equivalent fractions 

dy • (x+1) • (x-1) + y2 • x x2dy + xy2 - dy ——————————————————————————— = ——————————————— x x

Equation at the end of step  4  :

(x2dy + xy2 - dy) (x2d + dy2) - ————————————————— = 0 x

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a fraction from a whole 

Rewrite the whole as a fraction using  x  as the denominator :

x2d + dy2 (x2d + dy2) • x x2d + dy2 = ————————— = ——————————————— 1 x

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   x2d + dy2  =   d • (x2 + y2) 

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   x2dy + xy2 - dy  =   y • (x2d + xy - d) 

Trying to factor a multi variable polynomial :

 7.2    Factoring    x2d + xy - d 

Try to factor this multi-variable trinomial using trial and error 

 Factorization fails

Adding fractions that have a common denominator :

 7.3       Adding up the two equivalent fractions 

d • (x2+y2) • x - (y • (x2d+xy-d)) x3d - x2dy + xdy2 - xy2 + dy —————————————————————————————————— = ———————————————————————————— x x

Equation at the end of step  7  :

x3d - x2dy + xdy2 - xy2 + dy ———————————————————————————— = 0 x

Step  8  :

When a fraction equals zero :

 8.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

x3d-x2dy+xdy2-xy2+dy ———————————————————— • x = 0 • x x

Now, on the left hand side, the  x  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   x3d-x2dy+xdy2-xy2+dy  = 0

Solving a Single Variable Equation :

 8.2     Solve   x3d-x2dy+xdy2-xy2+dy  = 0 

In this type of equations, having more than one variable (unknown), you have to specify for which variable you want the equation solved.

We shall not handle this type of equations at this time.

Answer 2

a^2(xdy_ydx)÷x^2+y^2