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## Homework Statement

Solve y''=1+(y')

^{2}in two ways, for x is missing and y is missing.

## Homework Equations

Integration, and reduction of order.

## The Attempt at a Solution

First method: (this is correct)

y''=1+(y')

^{2}let y'=p and y''=p'

p'=1+p

^{2}

p'/(1+p

^{2})=1

∫p'/(1+p

^{2})dx=∫1dx

arctan(p)=x+c

solve for p: p=tan(x+c)

Substitute back for p=y'

y'=tan(x+c)

y=-ln(cos(x+c))+d

Second method: (I can't get it to come out as same answer from first method)

y''=1+(y')

^{2}let y'=p and y''=p*(dp/dy)

p*(dp/dy)=1+p

^{2}

From here I can already tell i'm not going to get y=-ln(cos(x+c))+d but this is the method I am told to use..

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