Thread: existence and uniqueness of nth-order IVP solutions

1. existence and uniqueness of nth-order IVP solutions

Hi guys. I'm given the following:

Determine the existence and uniqueness of the solutions of the initial value problem (IVP)

$\displaystyle y'''=(\sin t)e^y+y^2+(y')^{4/3}$,

$\displaystyle y(t_0)=a_0$, $\displaystyle y'(t_0)=a_1$, $\displaystyle y''(t_0)=a_2$.

I really don't know where to begin on this one. Any help would be much appreciated!

2. Let y0=y, y1= y', y2= y''. They y'''= y2' so the equation says y1'= sin(t)e^{y0}+ (y0)^2+ (y1)^{4/3} and y0'= y1. You can write those two equations as the single matrix equation
$\displaystyle \begin{bmatrix}y0 \\ y1\\ y2 \end{bmatrix}'= \begin{bmatrix}y1 \\ y2 \\ sin(t)e^{y_0}+ (y_0)^2+ (y_1)^{4/3}\end{bmatrix}$
with initial condition $\displaystyle \begin{bmatrix}y0(t_0) \\ y1(t_0) \\ y2(t_0)\end{bmatrix}=\begin{bmatrix}a_0 \\ a_1 \\ a_2\end{bmatrix}$

Now do you know the conditions for existence and uniqueness of solutions to a first order equation? Apply them to this equation.

3. Thanks for the starting point, but I'm still pretty lost. I've never done an exercise like this before, so it's giving me some real trouble!

We have the IVP $\displaystyle y'=f(y,t)$, where $\displaystyle y=(y_0,y_1,y_2)^T$. If it were just a basic real-valued function (instead of a vector-valued function), then I'd check for local Lipschitz continuity, maybe by taking the partial derivative $\displaystyle f_y(y,t)$, and seeing whether or not it is bounded, or else straight from the definition. But I have no idea how to look for Lipschitz continuity in a vector-valued function.

Thanks again for the help so far!