- #1
mr.tea
- 102
- 12
Hi,
I am learning ODE and I have some problems that confuse me.
In the textbook I am reading, it explains that if we have a separable ODE: ##x'=h(t)g(x(t))##
then ##x=k## is the only constant solution iff ##x## is a root of ##g##.
Moreover, it says "all other non-constant solutions are separated by the straight line x=k".
First, why do we do this separation between finding constant and non-constant solutions?
Second, I don't understand the quoted sentence. why is that?
Third, there is an example of finding a solution to the initial value problem ##x'=2tx^3## and ##x(0)=1##. They say that the only constant solution is ##x \equiv 0##, and
"Therefore if ##x(t)## is a solution such that ##x(0)=1##, then, by uniqueness, ##x(t)## cannot assume the value 0 anywhere. Since ##x(0) =1 >0##, we infer that the solution is always positive."
But how can ##x \equiv 0## be a constant solution if the solution should satisfy ##x(0)=1##, and how they got that the solution should be positive?
I am really confused and need some help with this.
Thank you.
I am learning ODE and I have some problems that confuse me.
In the textbook I am reading, it explains that if we have a separable ODE: ##x'=h(t)g(x(t))##
then ##x=k## is the only constant solution iff ##x## is a root of ##g##.
Moreover, it says "all other non-constant solutions are separated by the straight line x=k".
First, why do we do this separation between finding constant and non-constant solutions?
Second, I don't understand the quoted sentence. why is that?
Third, there is an example of finding a solution to the initial value problem ##x'=2tx^3## and ##x(0)=1##. They say that the only constant solution is ##x \equiv 0##, and
"Therefore if ##x(t)## is a solution such that ##x(0)=1##, then, by uniqueness, ##x(t)## cannot assume the value 0 anywhere. Since ##x(0) =1 >0##, we infer that the solution is always positive."
But how can ##x \equiv 0## be a constant solution if the solution should satisfy ##x(0)=1##, and how they got that the solution should be positive?
I am really confused and need some help with this.
Thank you.