"Consider the equation y3−32x2=1." was answered by students like you

Lipossig

Answered question

2021-03-31

Consider the equation

y^{3} - \frac{3}{2} x^{2} = 1

jlo2niT

Skilled2021-04-02Added 96 answers

Step 1
Consider the equation

y^{3} - \frac{3}{2} x^{2} = 1

Let us find the second derivative implicitly.
Step 2
Differentiate each term with respect to x

\frac{d}{d x} (y^{3} - \frac{3}{2} x^{2}) = \frac{d}{d x}

...(1)

\frac{d}{d x} (y^{3}) - \frac{3}{2} \frac{d}{d x} (x^{2}) = 0

3 y^{2} \frac{d y}{d x} - 3 x = 0

\frac{d y}{d x} = \frac{x}{y^{2}} \to (1)

Step 3
Differentiate second time with respect to x

\frac{d^{2} y}{{d x}^{2}} = \frac{d}{d x} (\frac{x}{y^{2}})

= \frac{\frac{d}{d x} (x) y^{2} - \frac{d}{d x} (y^{2} x)}{{(y^{2})}^{2}}

= \frac{1 \cdot y^{2} - 2 y \frac{d y}{d x} x}{y^{4}}

= \frac{y - 2 x \frac{d y}{d x}}{y^{3}}

Step 4
From equation (1) we have

\frac{d y}{d x} = \frac{x}{y^{2}}

. Substituting this we get

\frac{d^{2} y}{{d x}^{2}} = \frac{y - 2 x (\frac{x}{y^{2}})}{y^{3}}

= \frac{y^{3} - 2 x^{2}}{y^{5}}

Jeffrey Jordon

Expert2022-01-31Added 2605 answers

Answer is given below (on video)

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