- #1
BobbyBear
- 162
- 1
Consider the IVP:
[tex]
\left. \begin{array}{l}
\frac {dy} {dx} = f(x,y) \\
y( x_{0} ) = y_{0}
\end{array} \right\} \mbox{ze IVP :p}
[/tex]
Hypothesis:
[tex] f(x,y)\subset C^\infty_{x,y}(D)\; \; / \; \;(x_0,y_0)\in D [/tex]
[Note that this condition automatically satisfies the hypotheses of the Existence and Uniqueness Theorem (ie, [tex] f \in C_{x,y}(D)[/tex] and [tex]f \in L_y(D)[/tex] ([tex] L [/tex]=Lipschitzian)), hence we know that a unique solution to the IVP does indeed exist at least in a certain interval centred around [tex] x_0, \; x \in |x-x_0| \leq h [/tex]].
So then, let [tex]y(x)[/tex] be the solution to the IVP (at least in that interval [tex]x \in |x-x_0| \leq h [/tex]). Then we know that in that interval the ode is satisfied, and so,
[tex]\frac {dy} {dx} = f(x,y) \; \; \rightarrow \; \; y'(x_0)=f(x_0,y(x_0))=f(x_0,y_0)[/tex]
[tex]\frac {d^2y} {dx^2} = \frac {d} {dx} f(x,y)= \frac {\partial f} {\partial x} \frac {dx} {dx} \; + \; \frac {\partial f} {\partial y} \frac {dy} {dx} \\
\indent \rightarrow \; \; y''(x_0)= \frac {\partial f} {\partial x}|_{(x_0,y(x_0))} \; + \; \frac {\partial f} {\partial y}|_{(x_0,y(x_0))}\cdot f(x_0,y_0) [/tex]
[tex]\frac {d^3y} {dx^3} = \frac {d} {dx} (\frac {d^2y} {dx^2}) = \frac {\partial ^2f} {\partial x^2} \frac {dx} {dx} \; + \; \frac {\partial ^2f} {\partial y^2} \frac {dy} {dx} \; + \; \frac {\partial^2 f} {\partial x \partial y } \frac {dy} {dx} \; + \; \frac {\partial ^2f} {\partial y^2}(\frac {dy} {dx})^2 \; + \; \frac {\partial f} {\partial y} \frac {d^2y} {dx^2}\\ = \frac {\partial ^2f} {\partial x^2} \; + \; f\cdot \frac {\partial ^2f} {\partial y^2} \; + \; \frac {\partial f} {\partial y} (\frac {\partial f} {\partial x} \; + \; f\cdot \frac {\partial f} {\partial y}) \; + \; f\cdot \frac {\partial^2 f} {\partial x \partial y } \; + \; f^2 \frac {\partial ^2f} {\partial y^2} \\ \;
\indent \rightarrow \; \; [/tex] get [tex]y'''(x_0)[/tex] -wipes brow-
[Note that for the derivatives of y to exist we need the derivatives of all orders of [tex]f [/tex] to exist, at least in the small region around [tex](x_0, y_0)[/tex], which justifies the hypothesis of the method].
So! We can then construct the Taylor series:
[tex]y(x_0)+y'(x_0)\cdot (x-x_0) + y''(x_0)\cdot \frac {(x-x_0)^2}{2!} + . . . [/tex]
BUT!
1) How do you know that the Taylor series converges around x0 ? (Do all Taylor series converge in some interval around x0? :P )
2) Okay so assuming the Taylor series converges in some interval around x0, how do we know that it is equal to the function [tex]y(x)[/tex] solution of the IVP? That [tex]y(x) \in C^\infty _x [/tex] does NOT mean that [tex]y(x)[/tex] is analytic! :>
PALEEZE HELP! 0:
[tex]
\left. \begin{array}{l}
\frac {dy} {dx} = f(x,y) \\
y( x_{0} ) = y_{0}
\end{array} \right\} \mbox{ze IVP :p}
[/tex]
Hypothesis:
[tex] f(x,y)\subset C^\infty_{x,y}(D)\; \; / \; \;(x_0,y_0)\in D [/tex]
[Note that this condition automatically satisfies the hypotheses of the Existence and Uniqueness Theorem (ie, [tex] f \in C_{x,y}(D)[/tex] and [tex]f \in L_y(D)[/tex] ([tex] L [/tex]=Lipschitzian)), hence we know that a unique solution to the IVP does indeed exist at least in a certain interval centred around [tex] x_0, \; x \in |x-x_0| \leq h [/tex]].
So then, let [tex]y(x)[/tex] be the solution to the IVP (at least in that interval [tex]x \in |x-x_0| \leq h [/tex]). Then we know that in that interval the ode is satisfied, and so,
[tex]\frac {dy} {dx} = f(x,y) \; \; \rightarrow \; \; y'(x_0)=f(x_0,y(x_0))=f(x_0,y_0)[/tex]
[tex]\frac {d^2y} {dx^2} = \frac {d} {dx} f(x,y)= \frac {\partial f} {\partial x} \frac {dx} {dx} \; + \; \frac {\partial f} {\partial y} \frac {dy} {dx} \\
\indent \rightarrow \; \; y''(x_0)= \frac {\partial f} {\partial x}|_{(x_0,y(x_0))} \; + \; \frac {\partial f} {\partial y}|_{(x_0,y(x_0))}\cdot f(x_0,y_0) [/tex]
[tex]\frac {d^3y} {dx^3} = \frac {d} {dx} (\frac {d^2y} {dx^2}) = \frac {\partial ^2f} {\partial x^2} \frac {dx} {dx} \; + \; \frac {\partial ^2f} {\partial y^2} \frac {dy} {dx} \; + \; \frac {\partial^2 f} {\partial x \partial y } \frac {dy} {dx} \; + \; \frac {\partial ^2f} {\partial y^2}(\frac {dy} {dx})^2 \; + \; \frac {\partial f} {\partial y} \frac {d^2y} {dx^2}\\ = \frac {\partial ^2f} {\partial x^2} \; + \; f\cdot \frac {\partial ^2f} {\partial y^2} \; + \; \frac {\partial f} {\partial y} (\frac {\partial f} {\partial x} \; + \; f\cdot \frac {\partial f} {\partial y}) \; + \; f\cdot \frac {\partial^2 f} {\partial x \partial y } \; + \; f^2 \frac {\partial ^2f} {\partial y^2} \\ \;
\indent \rightarrow \; \; [/tex] get [tex]y'''(x_0)[/tex] -wipes brow-
[Note that for the derivatives of y to exist we need the derivatives of all orders of [tex]f [/tex] to exist, at least in the small region around [tex](x_0, y_0)[/tex], which justifies the hypothesis of the method].
So! We can then construct the Taylor series:
[tex]y(x_0)+y'(x_0)\cdot (x-x_0) + y''(x_0)\cdot \frac {(x-x_0)^2}{2!} + . . . [/tex]
BUT!
1) How do you know that the Taylor series converges around x0 ? (Do all Taylor series converge in some interval around x0? :P )
2) Okay so assuming the Taylor series converges in some interval around x0, how do we know that it is equal to the function [tex]y(x)[/tex] solution of the IVP? That [tex]y(x) \in C^\infty _x [/tex] does NOT mean that [tex]y(x)[/tex] is analytic! :>
PALEEZE HELP! 0: