Parametric function differentiation problem x = a cos &#x2061;<!-- ⁡ --> g (

Santino Bautista

Santino Bautista

Answered question

2022-06-13

Parametric function differentiation problem
x = a cos g ( t )  and  y = b sin g ( t )
x y 2 d 2 y d x 2 = b 2 d y d x

Answer & Explanation

Angelo Murray

Angelo Murray

Beginner2022-06-14Added 23 answers

Step 1
Evaluate d y d t and d x d t . Then,
d y d x = d y / d t d x / d t
d 2 y d x 2 = d d x ( d y d x )
Now simply substitute x, y into d y d x , , and d 2 y d x and then all of these into your second equation.
Yesenia Sherman

Yesenia Sherman

Beginner2022-06-15Added 5 answers

Step 1
It is noticed that eliminating g(t) right at the start we need only to deal with two differentiations on a standard ellipse without parameter t:
(1) x 2 a 2 + y 2 b 2 = 1
This may not be strictly in line with the aim of the exercise. Also it was on for 6 years.
So however to proceed denote 's with resp to x,
(2) x a 2 + y y b 2 = 0 ; y = x b 2 y a 2
(3) 1 a 2 + y y + y y 2 b 2 = 0 ; y = b 2 / a 2 y 2 y
Plug in 2) and 3) into the given equation
(4) x y 2 y b 2 y = 0
and simplify LHS:
(5) x a 2 y ( x 2 a 2 + y 2 b 2 1 )   ,
that vanishes.

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