I need help figuring out why the fisr order ODE : $x'(t)=x^2(t)$ with the initial condition $x(0)=x_{0}\neq 0$ doesn't have a solution defined for all t.
Does anyone have an idea ?
I need help figuring out why the fisr order ODE : $x'(t)=x^2(t)$ with the initial condition $x(0)=x_{0}\neq 0$ doesn't have a solution defined for all t.
Does anyone have an idea ?
$x'(t)=x^2(t)$
$\frac{dx}{dt}=x^2$
$\frac{dx}{x^2}=dt$
$\int\frac{dx}{x^2}=\int dt$
$\frac{-1}{x}=t+c$
$x(t)=\frac{-1}{t+c}$
if $x(0)=x_0$ then
$x(0)=\frac{-1}{0+c}=\frac{-1}{c}=x_0$ ($x_0\neq 0$)
$c=\frac{-1}{x_0}$
$x(t)=\frac{-1}{t+(\frac{-1}{x_0})}=\frac{x_0}{1-x_0t}$
Let's find the values that $x(t)$ is not defined:
$1-x_0t=0$
$t=\frac{1}{x_0}$
if $t=\frac{1}{x_0}$ then $x(t)$ is not defined. $x(t)$ is defined for all other values except $t=\frac{1}{x_0}$ ($x_0\neq 0$).
Nobody has said that any initial value problem you cook up will have a solution for all $t$ – in the same vein that you cannot expect a real solution for all equations of the form $x^2=c\in{\mathbb R}$.
On the positive side you can say the following: When the hypotheses of the existence and uniqueness theorem are satisfied throughout the domain $\Omega\subset{\mathbb R}^2$ of the given ODE $x'=f(x,t)$ then the graph of any "complete" solution will finally leave any given compact $K\subset\Omega$.
In the case at hand the graph of the solution $y(t)=x_0/(1-t x_0)$ will go off to $\infty$ when $t x_0\to 1-$.