I'm studying Numerical Analysis and the first example given in Suli and Mayers book on the section on initial value problems is the following -
Consider the differential equation $y' = |y|^{\alpha}$, subject to the initial condition $y(0) = 0$, where $\alpha$ is a fixed real number, $\alpha \in (0,1)$.
It is a simple matter to verify that, for any nonnegative real number $c$,
$y_c(x) = (1 - \alpha)^{\frac{1}{1 - \alpha}}(x - c)^{\frac{1}{1 - > \alpha}}$ when $c<=x< \infty$
$y_c(x) = 0$ when $0<=x< c$
is a solution to the initial value problem on the interval $[0, \infty)$.
I have not studied differential equations before so it is not a 'simple matter' for me to verify the solution to this equation. Can someone explain to me what is going on and how that $y_c(x)$ is a solution to the initial value problem? I don't see where the $x$ is 'coming from' for example...I see no $x$ in $y' = |y|^{\alpha}$.