Solved: "Second derivatives Find d2ydx2. e2y+x=y" - Plainmath

Cabiolab

Answered question

2021-10-14

Second derivatives Find

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

.

e^{2 y} + x = y

Answer & Explanation

faldduE

Skilled2021-10-15Added 109 answers

Step 1
Given equation is

e^{2 y} + x = y

.
Compute the derivative with respect to x.

\frac{d y}{d x} = 2 e^{2 y} \frac{d y}{d x} + 1

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

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

Step 2
Again compute the derivative with respect to x.

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

= \frac{4 e^{2 y}}{(1 - 2 e^{2 y})} \frac{d y}{d x}

= (\frac{4 e^{2 y}}{{(1 - 2 e^{2 y})}^{2}}) (\frac{1}{1 - 2 e^{2 y}}) [\frac{d y}{d x} = \frac{1}{1 - 2 e^{2 y}}]

= \frac{4 e^{2 y}}{{(1 - 2 e^{2 y})}^{3}}

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