Existence-uniqueness: Difference between revisions
(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...") |
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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 | [[Category:Differential Equations]] | ||
Revision as of 19:29, 17 May 2024
Consider an IVP for a general first order scalar ODE.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} y' = f(t,y)\\ y(t_0) = y_0 \end{cases} }
The existence-uniqueness theorem theorem states that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t,y)} and its derivative w.r.t. y is continuous in some rectangle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a<t<b} , about Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (t_0, y_0)} , then there exists a unique solution of the IVP defined for some time interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a'<t<b'} about Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_0} .
Notes
The time interval may be much shorter than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a<t<b} . 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t,y)} is the slope. Consider the vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1,f(t,y))} . This vector would be tangential to the movement of the curve.
