Find $(d^2y)/(dx^2)∣_[(x,y)=(2,1)]$ if y is a differentiable function of $x$ satisfying the equation $x^3+2y^3 = 5xy$ ? Plot it?

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Find $(d^2y)/(dx^2)∣_[(x,y)=(2,1)]$ if y is a differentiable function of $x$ satisfying the equation $x^3+2y^3 = 5xy$ ? Plot it?

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Answer 1 · 2016-05-29T03:27:31

$125/16$

Explanation:

Taking the derivative of $x^3+2y^3=5xy$ with respect to $x$ , we see that

$3x^2+6y^2dy/dx=5y+5xdy/dx$

Rearranging and solving for $dy/dx$ shows that

$dy/dx=(5y-3x^2)/(6y^2-5x)$

Note that

$(dy)/(dx)∣_[(x,y)=(2,1)]=(5-12)/(6-10)=7/4$

Differentiating once more:

$(d^2y)/dx^2=((5ydy/dx-6x)(6y^2-5x)-(12ydy/dx-5)(5y-3x^2))/(6y^2-5x)^2$

When evaluating this at $(2,1)$ , each instance of $dy/dx$ becomes $7/4$ .

$(d^2y)/dx^2|_[(x,y)=(2,1)]=((5(7/4)-12)(6-10)-(12(7/4)-5)(5-12))/(6-10)^2$

$(d^2y)/dx^2|_[(x,y)=(2,1)]=((-13/4)(-4)-16(-7))/16$

$(d^2y)/dx^2|_[(x,y)=(2,1)]=(13+112)/16=125/16$

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Find $(d^2y)/(dx^2)∣_[(x,y)=(2,1)]$ if y is a differentiable function of $x$ satisfying the equation $x^3+2y^3 = 5xy$ ? Plot it?

1 Answer

Explanation:

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Find (d^2y)/(dx^2)∣_[(x,y)=(2,1)](d^2y)/(dx^2)∣_[(x,y)=(2,1)] if y is a differentiable function of xx satisfying the equation x^3+2y^3 = 5xyx^3+2y^3 = 5xy? Plot it?

1 Answer

Explanation:

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Find $(d^2y)/(dx^2)∣_[(x,y)=(2,1)]$ if y is a differentiable function of $x$ satisfying the equation $x^3+2y^3 = 5xy$ ? Plot it?