Existence-uniqueness - Revision history

Rice at 00:23, 18 May 2024

2024-05-18T00:23:34Z

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 Consider an IVP for a general first order scalar [[Ordinary differential equation|ODE]].
@@ Line 8: / Line 25: @@
 </math>
-The '''existence-uniqueness theorem''' theorem states that if <math>f(t,y)</math> and <math>\frac{\partial f}{\partial y}</math> is continuous in some rectangle <math>a<t<b</math>, <math>c<y<d</math> about <math>(t_0, y_0)</math>, then there exists a unique solution of the IVP defined for some time interval <math>a'<t<b'</math> about <math>t_0</math>.
+The theorem states that if <math>f(t,y)</math> and <math>\frac{\partial f}{\partial y}</math> is continuous in some rectangle <math>a<t<b</math>, <math>c<y<d</math> about <math>(t_0, y_0)</math>, then there exists a unique solution of the IVP defined for some time interval <math>a'<t<b'</math> about <math>t_0</math>.
 = Notes =
@@ Line 17: / Line 34: @@
 = Geometric intuition =
 Geometrically, <math>f(t,y)</math> is the slope. Consider the vector <math>(1,f(t,y))</math>. This vector would be tangential to the movement of the curve.

Rice at 21:05, 17 May 2024

2024-05-17T21:05:41Z

← Older revision		Revision as of 21:05, 17 May 2024
Line 8:		Line 8:
	</math>		</math>

	The '''existence-uniqueness theorem''' theorem states that if <math>f(t,y)</math> and ~~its derivative w.r.t.~~ y is continuous in some rectangle <math>a<t<b</math>, <math>c<y<d</math> about <math>(t_0, y_0)</math>, then there exists a unique solution of the IVP defined for some time interval <math>a'<t<b'</math> about <math>t_0</math>.		The '''existence-uniqueness theorem''' theorem states that if <math>f(t,y)</math> and <math>\frac{\partial f}{\partial y}</math> is continuous in some rectangle <math>a<t<b</math>, <math>c<y<d</math> about <math>(t_0, y_0)</math>, then there exists a unique solution of the IVP defined for some time interval <math>a'<t<b'</math> about <math>t_0</math>.

	= Notes =		= Notes =
	The time interval may be much shorter than <math>a<t<b</math>. The solution gives ''no information'' on the time interval. This means that the solution can go off to undefined values even when the inputs are defined.		The time interval may be much shorter than <math>a<t<b</math>. The solution gives ''no information'' on the time interval. This means that the solution can go off to undefined values even when the inputs are defined.

			Differentiability is important to ensure that the solution is unique.

	= Geometric intuition =		= Geometric intuition =

Rice at 19:29, 17 May 2024

2024-05-17T19:29:05Z

← Older revision		Revision as of 19:29, 17 May 2024
Line 16:		Line 16:
	Geometrically, <math>f(t,y)</math> is the slope. Consider the vector <math>(1,f(t,y))</math>. This vector would be tangential to the movement of the curve.		Geometrically, <math>f(t,y)</math> is the slope. Consider the vector <math>(1,f(t,y))</math>. This vector would be tangential to the movement of the curve.

	[[Category:Differential ~~Equation~~]]		[[Category:Differential Equations]]

Rice: Created page with "Consider an IVP for a general first order scalar ODE. $\begin{cases} y' = f(t,y)\\ y(t_0) = y_0 \end{cases}$ The '''existence-uniqueness theorem''' theorem states that if $f(t,y)$ and its derivative w.r.t. y is continuous in some rectangle $a, c about (t_0, y_0), then there exists a unique solution of the IVP defined for some time interval a' about..."$

2024-04-26T22:27:28Z

Created page with "Consider an IVP for a general first order scalar ODE. <math> \begin{cases} y' = f(t,y)\\ y(t_0) = y_0 \end{cases} </math> The '''existence-uniqueness theorem''' theorem states that if <math>f(t,y)</math> and its derivative w.r.t. y is continuous in some rectangle <math>a<t<b</math>, <math>c<y<d</math> about <math>(t_0, y_0)</math>, then there exists a unique solution of the IVP defined for some time interval <math>a'<t<b'</math> about..."

New page

Consider an IVP for a general first order scalar [[Ordinary differential equation|ODE]].

<math>
\begin{cases}
y' = f(t,y)\\
y(t_0) = y_0
\end{cases}
</math>

The '''existence-uniqueness theorem''' theorem states that if <math>f(t,y)</math> and its derivative w.r.t. y is continuous in some rectangle <math>a<t<b</math>, <math>c<y<d</math> about <math>(t_0, y_0)</math>, then there exists a unique solution of the IVP defined for some time interval <math>a'<t<b'</math> about <math>t_0</math>.

= Notes =
The time interval may be much shorter than <math>a<t<b</math>. The solution gives ''no information'' on the time interval. This means that the solution can go off to undefined values even when the inputs are defined.

= Geometric intuition =
Geometrically, <math>f(t,y)</math> is the slope. Consider the vector <math>(1,f(t,y))</math>. This vector would be tangential to the movement of the curve.

[[Category:Differential Equation]]