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Gummy Bear
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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+p2
p'/(1+p2)=1
∫p'/(1+p2)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+p2
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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