Does $y' = t − y^2$ have solutions defined on some interval $(a, \infty)$?

Question

Consider the equation $y' = t − y^2$ . Does it have solutions defined on some interval $(a, \infty)$?

Any help is appreciated!

Answer 1

The solutions are $$y = \dfrac{c_1 \text{Ai}'(t) + c_2 \text{Bi}'(t)}{c_1 \text{Ai}(t) + c_2 \text{Bi}(t)}$$ where $\text{Ai}$ and $\text{Bi}$ are Airy wave functions and $c_1$ and $c_2$ are not both $0$. For any $c_1$ and $c_2$, these are indeed defined on some $(a,\infty)$, as the denominator is $0$ for only finitely many $t > 0$ and the function is analytic otherwise.

You can also look at this using a phase plane analysis. Any solution that starts in or reaches the region $y^2 < t$ must stay in that region, and then is defined for all future time.

Answer 2

$$y'=t-y^2$$ This is a Riccati ODE. When no particular solution can be guessed, the usual method to solve it is to convert the ODE to a linear second order ODE, because second order linear ODEs are usually easier to solve that non-linear first order ODEs.

Let $\quad y(t)=\frac{u'(t)}{u(t)}$ $$y'=\frac{u''}{u}-\frac{(u')^2}{u^2}=t-\left(\frac{u'}{u}\right)^2$$ $$u''-t_:u=0$$

This is an Airy ODE. The solutions are the Airy functions : http://mathworld.wolfram.com/AiryFunctions.html $$u(t)=c_1\text{Ai}(t)+c_2\text{Bi}(t)$$ For the derivatives Ai'$(t)$ and Bi'$(t)$ see : http://functions.wolfram.com/Bessel-TypeFunctions/AiryAiPrime/ and http://functions.wolfram.com/Bessel-TypeFunctions/AiryBiPrime/

$$y(t)=\frac{c_1\text{Ai'}(t)+c_2\text{Bi'}(t)}{c_1\text{Ai}(t)+c_2\text{Bi}(t)}$$

The Airy function are related to some Bessel functions. So, the result can be written as well in terms of Bessel functions.

Answer 3

You get for $0\le t$ $$ -y^2\le y'\le t $$ which can both be integrated to $$ \frac{y(0)}{1+y(0)t}\le y(t)\le y(0)+\frac12t^2. $$ This means that for $y(0)>0$ the solution stays bounded over all finite intervals $[0,b]$ and can thus be extended to a solution over $[0,\infty)$. By local existence the maximal solution will exist on some interval $(a,\infty)$ with $a<0$.

Your question did not indicate that a statement for arbitrary initial conditions was sought.