Y^3+XY+X^2=0 and I was suppose to find dY/dX and substitute the value 10 for X after differentiating. Weirdly enough, this equation does not let me get rid of the variable Y after solving and hence I couldn't find a valid answer. Would request you to help me approach the problem. Also, I would like to do so without finding the roots of Y and solving, as someone pointed out earlier in class.

hikstac0 2022-09-30 Answered
Y 3 + X Y + X 2 = 0
and I was suppose to find dY/dX and substitute the value 10 for X after differentiating. Weirdly enough, this equation does not let me get rid of the variable Y after solving and hence I couldn't find a valid answer.

Would request you to help me approach the problem. Also, I would like to do so without finding the roots of Y and solving, as someone pointed out earlier in class.
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Answers (2)

xgirlrogueim
Answered 2022-10-01 Author has 13 answers
Let F = y 3 + x y + x 2 . Then
F x = y + 2 x
F y = 3 y 2 + x
d y d x = F x F y = y + 2 x 3 y 2 + x
We know x = 10 and we need the corresponding y which is the solution of y 3 + 10 y + 100 = 0 which is cubic equation that you can solve using Cardano. This equation has a single real root given by
y = 10 ( 2055 45 ) 3 3 2 / 3 10 2 / 3 3 ( 2055 45 ) 3 3.93003
If you do not use Cardano, only a numerical method could be used. The simplest would be Newton, which, starting from a "reason able" guess y 0 will update it according to
y n + 1 = y n f ( y n ) f ( y n )
with f ( y ) = y 3 + 10 y + 100. By inspection, you could see that there is a root close to −4; so, let us select y 0 = 4. Soo, Newton iterates will be −3.93103,−3.93003.
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planhetkk
Answered 2022-10-02 Author has 2 answers
Y 3 + X Y + X 2 = 0
3 Y 2 d Y + Y d X + X d Y + 2 X d X = 0
( 3 Y 2 + X ) d Y = ( Y + 2 X ) d X
d Y d X = Y + 2 X 3 Y 2 + X
Are you sure that you need more ? If yes, you will have to solve Y 3 + X Y + X 2 = 0 for Y and put it back into the above formula.
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