Question

why is y=0 also a general solution for this example
http://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-i-first-order-differential-equations/basic-de-and-separable-equations/MIT18_03SCF11_s1_5text.pdf
??

Answer 1

Substitute \[y=0\] in the differential equation \[\frac{ dy }{ dx } = y^{2}\] on LHS:
\[\frac{ dy }{ dx } =\frac{ d }{ dx }(0) = 0\] and on RHS \[y^{2} = 0^{2} = 0\] Since both sides are 0, y=0 is a solution to the differential equation, and is part of the general solution. As a side note- this is a lost solution i.e. by separating variables for solving the differential equation, the solution y=0 was lost. So always be on watch for lost solutions :D