Here is the question:
y'' = (y')^2 (Assume y and y' are positive)
I looked at the question and didn't know how to start it, never seen a question in this form. Help please!!!
Just put
y'=f.
The equation then becomes
$\displaystyle
\frac{df}{dx}=f^2(x)
$
Separate and integrate to get
$\displaystyle
f^{-2}df=dx
$
and
$\displaystyle
-f^{-1}=x+a
$,
Hence
$\displaystyle
f(x)=-\frac{1}{x+a}.
$
Now,
$\displaystyle
y'=-\frac{1}{x+a}
$
Integrate to get the final answer
$\displaystyle
y(x)=-\ln(|x+a|)+b
$