Show IVP solution exactness in given interval

Question

How do I show that solution of $ y' = e^{-t^2}+y^2$, $y(0)=0$ is exact in $0\leq t\leq \frac{1}{2}$?

Answer 1

I believe you mean existence - not exactness. In any case, you have a Riccati equation, which is linearlized by the change of variable $$y=-\frac{u'}{u}, \tag{1} $$ into $$u''=-\mathrm{e}^{-t^2} u.$$ The initial conditions are transformed into $$u(0)=1,u'(0)=0.$$ (In fact, the scaling symmetry of $(1)$ allows you to take any nonzero value for $u(0)$.)

Singularities will arise in $y$ precisely when $u=0$. Suppose we restrict ourselves to a time interval $t \in [0,T]$, in which $u$ is positive. We then have the following differential inequality over that interval $$u'' \geq -u ,$$ and the theory of linear differential inequalities implies that upon integration, $$u(t) \geq \cos(t). $$

Hence $u$ is positive at least in the interval $[0,\pi/2)$, and correspondingly, $y$ is smooth there.