uy=F(x,y,ux) unique solution: uy=F(x,y,ux) u(x,0)=g(x)

E

express78

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For the IVP
[FONT=MathJax_Math-italic]u[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]F[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]u[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Math-italic]
u[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]g[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main])[/FONT]
​
where F and g is three times continuously differentiable. I have tried to show that the problem has a unique solution.
How i can show a unique solution by charactristic? or by other method.
 
express78 said:
For the IVP
[FONT=MathJax_Math-italic]u[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]F[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]u[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Math-italic]
u[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]g[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main])[/FONT]
​
where F and g is three times continuously differentiable. I have tried to show that the problem has a unique solution.
How i can show a unique solution by charactristic? or by other method.
Click to expand...
does uy in your post mean:

\(\displaystyle \displaystyle{uy \ = \ u_y \ = \ \frac{d}{dy} \left [u(x,y) \right ]_{x=constant}}\)
 
express78 said:
For the IVP
[FONT=MathJax_Math-italic]u[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]F[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]u[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Math-italic]
u[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math-italic]g[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main])[/FONT]
​
where F and g is three times continuously differentiable. I have tried to show that the problem has a unique solution.
How i can show a unique solution by charactristic? or by other method.
Click to expand...
When you reply with the previously-requested information, please include the instructions for this exercise, as well as a clear listing of your efforts thus-far to resolve this exercise. Thank you! ;)
 
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