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Post Closed as "Not suitable for this site" by Will Jagy, Andrey Rekalo, Denis Serre, Michael Renardy, Willie Wong
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Ricardo Andrade
Ricardo Andrade
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Finding Conditionsconditions to guarantee existence of solutions to IVP

Consider the following IVP: $x'=f(t,x)$ $\ $and $\ $and $x(0)=x_0$, where $x\in \mathbb{R^n}$$x\in \mathbb{R}^n$ and $t\in \mathbb{R}$.

Suppose that for all $(t,x)\in\mathbb{R^{n+1}}$:$(t,x)\in\mathbb{R}^{n+1}$, $|f(t,x)|\leq b(t) |x|^2$.

In order for the solution to the IVP to exist for all $t>=0$, what conditions should we impose on $b(t)$ and $x_0$?

I know that if the function $f: \mathbb{R^{n+1}}\to \mathbb{R^n}$$f: \mathbb{R}^{n+1}\to \mathbb{R}^n$ is continuous, then there is a theorem (Peano Local existence Theorem) that guarantees that the solution to the IVP exists, but I can't seem to find a way to prove the continuity of $f$. I wonder if anyone of you knows another way on how to solve this problem. Thanks!

Finding Conditions to guarantee existence of solutions to IVP

Consider the following IVP: $x'=f(t,x)$ $\ $and $\ $ $x(0)=x_0$ where $x\in \mathbb{R^n}$ and $t\in \mathbb{R}$.

Suppose that for all $(t,x)\in\mathbb{R^{n+1}}$: $|f(t,x)|\leq b(t) |x|^2$.

In order for the solution to the IVP to exist for all $t>=0$, what conditions should we impose on $b(t)$ and $x_0$?

I know that if the function $f: \mathbb{R^{n+1}}\to \mathbb{R^n}$ is continuous, then there is a theorem (Peano Local existence Theorem) that guarantees that the solution to the IVP exists, but I can't seem to find a way to prove the continuity of $f$. I wonder if anyone of you knows another way on how to solve this problem. Thanks!

Finding conditions to guarantee existence of solutions to IVP

Consider the following IVP: $x'=f(t,x)$ and $x(0)=x_0$, where $x\in \mathbb{R}^n$ and $t\in \mathbb{R}$.

Suppose that for all $(t,x)\in\mathbb{R}^{n+1}$, $|f(t,x)|\leq b(t) |x|^2$.

In order for the solution to the IVP to exist for all $t>=0$, what conditions should we impose on $b(t)$ and $x_0$?

I know that if the function $f: \mathbb{R}^{n+1}\to \mathbb{R}^n$ is continuous, then there is a theorem (Peano Local existence Theorem) that guarantees that the solution to the IVP exists, but I can't seem to find a way to prove the continuity of $f$. I wonder if anyone of you knows another way on how to solve this problem. Thanks!

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User267845467
User267845467
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Finding Conditions to guarantee existence of solutions to IVP

Consider the following IVP: $x'=f(t,x)$ $\ $and $\ $ $x(0)=x_0$ where $x\in \mathbb{R^n}$ and $t\in \mathbb{R}$.

Suppose that for all $(t,x)\in\mathbb{R^{n+1}}$: $|f(t,x)|\leq b(t) |x|^2$.

In order for the solution to the IVP to exist for all $t>=0$, what conditions should we impose on $b(t)$ and $x_0$?

I know that if the function $f: \mathbb{R^{n+1}}\to \mathbb{R^n}$ is continuous, then there is a theorem (Peano Local existence Theorem) that guarantees that the solution to the IVP exists, but I can't seem to find a way to prove the continuity of $f$. I wonder if anyone of you knows another way on how to solve this problem. Thanks!