Can anybody help me with the proof of Existence and Uniqueness Theorems.
heman said:Can anybody help me with the proof of Existence and Uniqueness Theorems.
saltydog said:A unique solution of this equation about the point [itex](x_0,y_0)[/itex] exists if f(x,y) and it's partial with respect to y is continuous about the point.
heman said:Thanks for Amazing Clarification Salty but my doubt is this that i want to know why the happening of continuity of F(x,y) and Fy(x,y) ensures the uniqueness.!!
HallsofIvy said:Actually, continuity of Fy(x,y) is not necessary. What is necessary for uniqueness is that F be "Lipschitz" in y. A function of one variable is Lipschitz (on a set) if and only if [itex]|f(x)-f(y)|\leC|x-y|[/itex].
Can you give me an online link on Differential eqns. which stresses on proof!
Actually my textbook Kreyzig sucks!
saltydog said:Heman, really I think only "text book in hand" is the best way to learn the proof. Go through each line carefully and make sure you understand every detail. Draw pictures, go back and review all the theorems that are quoted. Do examples.
Also, I have Kreyzig. It's a good book for Engineers but not for Mathematicians.
God I hope I don't get in trouble with the Engineers in here for saying that.