Question

How do you find `(d^2y)/(dx^2)` given `x^3+y^3=18xy`?

Answer 1

`y^('')(x) = (2 x y(x) (216 + x^3 - 18 x y(x) + y(x)^3))/(6 x - y(x)^2)^3`

Explanation:

Calling

`f(x,y) = x^3+y^3(x)-18x y(x) = 0`

#(df)/(dx)=3 x^2 - 18 y(x) - 18 x y'(x) + 3 y(x)^2 y'(x) = 0#

giving

`y'(x) = (x^2 - 6 y(x))/(6 x - y(x)^2)`

Now

#d/(dx)((df)/(dx))=6 x - 36 y'(x) + 6 y(x) y'(x)^2 - 18 x y^{''}(x)+ 3 y(x)^2 y^{''}(x)=0#

Solving for `y^('')(x)` we have

#y^('')(x) = (2 (x - 6 y'(x)+ y(x) y'(x)^2))/( 6 x - y(x)^2)#

and after substituting `y'(x)` we obtain finally

`y^('')(x) = (2 x y(x) (216 + x^3 - 18 x y(x) + y(x)^3))/(6 x - y(x)^2)^3`

Answer 2

`(d^2y)/(dx^2)=(2xy(y^3 + 216 - 18xy + x^3))/(6x-y^2)^3`

Explanation:

We first find the first derivative, paying careful attention to `y(x)` where we need to apply the chain rule:

`d/(dx) ( x^3 + y^3 = 18xy)`

`rarr 3x^2 + 3y^2(dy)/(dx) = 18y + 18x(dy)/(dx)`

Rearrange to obtain expression for `(dy)/(dx)`:

`(dy)/(dx) = (3x^2 - 18y)/(18x - 3y^2)" " color(red)("Equation 1")`

Simplify:

`(dy)/(dx) = (x^2-6y)/(6x-y^2)`

Differentiate again using quotient rule:

`(d^2y)/(dx^2) = ((2x-6(dy)/(dx))(6x-y^2) - (x^2-6y)(6 - 2y(dy)/(dx)))/(6x-y^2)^2`

`(d^2y)/(dx^2) = (color(blue)(12x^2) - 2xy^2 -36x(dy)/(dx) + color(orange)(6y^2(dy)/(dx))- color(blue)( 6x^2) + 2x^2y(dy)/(dx) + 36y -color(orange)( 12y^2(dy)/(dx)))/(6x-y^2)^2`

Some stuff cancels which is nice.

`(d^2y)/(dx^2) = (6x^2 - 2xy^2 - 36x(dy)/(dx) - 6y^2(dy)/(dx) + 2x^2y(dy)/(dx) + 36y)/(6x-y^2)^2`

Now we want to sub in equation 1:

`= (6x^2 - 2xy^2 - 36x((x^2-6y)/(6x-y^2)) - 6y^2((x^2-6y)/(6x-y^2)) + 2x^2y((x^2-6y)/(6x-y^2)) + 36y)/(6x-y^2)^2`

Make every term in the numerator over a common denominator:

`= (6x^2(6x-y^2)/(6x-y^2) - 2xy^2(6x-y^2)/(6x-y^2) - 36x((x^2-6y)/(6x-y^2)) - 6y^2((x^2-6y)/(6x-y^2)) + 2x^2y((x^2-6y)/(6x-y^2)) + 36y(6x-y^2)/(6x-y^2))/(6x-y^2)^2`

Can move this down to the full denominator:

`=(6x^2(6x-y^2) - 2xy^2(6x-y^2) - 36x(x^2-6y) - 6y^2(x^2-6y) + 2x^2y(x^2-6y) + 36y(6x-y^2))/(6x-y^2)^3`

Expanding:

`=(color(blue)(36 x^3)-6 x^2 y^2-12 x^2 y^2+2 x y^4-color(blue)(36 x^3)+216 x y-6 x^2 y^2+color(orange)(36 y^3)+2 x^4 y-12 x^2 y^2+216 x y-color(orange)(36 y^3))/(6x-y^2)^3`

Again, we can cancel a few things, marked in colour.

We now collect like terms to obtain:

`=(2xy^4 + 432xy - 36x^2y^2 + 2x^4y)/(6x-y^2)^3`

Factor to finally obtain that:

`(d^2y)/(dx^2)=(2xy(y^3 + 216 - 18xy + x^3))/(6x-y^2)^3`