Answer to Question #148900 in Calculus for eddi

Question #148900

(x-y)^3=A(x+y), prove that (2x+y)dy/dx = x + 2y


1
Expert's answer
2020-12-08T06:25:50-0500

If "x+y=0=>x=-y"


"(x+x)^3=0=>x=0=>y=0"

"x=-y=>\\dfrac{dy}{dx}=-1"

"(2(0)+0)(-1)=0+2(0), True"

If "x+y\\not=0"


"\\dfrac{(x-y)^3}{x+y}=A"

Differentiate both sides with respect to "x"


"\\dfrac{d}{dx}\\big(\\dfrac{(x-y)^3}{x+y}\\big)=\\dfrac{d}{dx}(A)"

Use the Chain Rule


"\\dfrac{3(x-y)^2(1-\\dfrac{dy}{dx})(x+y)-(1+\\dfrac{dy}{dx})(x-y)^3}{(x+y)^2}=0"

"(x-y)^2(3x+3y-3x\\dfrac{dy}{dx}-3y\\dfrac{dy}{dx}-x+y-x\\dfrac{dy}{dx}+y\\dfrac{dy}{dx})=0"

If "x=y," "\\dfrac{dy}{dx}=1"


"(2x+x)(1)=x+2x, True"

Or


"2x+4y-4x\\dfrac{dy}{dx}-2y\\dfrac{dy}{dx}=0"

"(2x+y)\\dfrac{dy}{dx}=x+2y, True"

Therefore


"(2x+y)\\dfrac{dy}{dx}=x+2y, \\text {True, for each } x, y"


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS