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Write the single nth order equation u^{{(n)}}=g(u,u',...,u^{{(n-1)}},t)\, as a first-order system.


Let x_{1}=u,x_{2}=u',...,x_{n}=u^{{(n-1)}}\,

Now form the first order system

x_{1}'=x_{2}\,
x_{2}'=x_{3}\,
...\,
x_{{n-1}}'=x_{n}\,
x_{n}'=g(x_{1},x_{2},...,x_{n},t)\,

With the following initial conditions

u(t_{0})=u,u'(t_{0})=u_{2},...,u^{{(n-1)}}(t_{0})=u_{n}\,

Let {\vec  x}={\begin{bmatrix}x_{1}\\x_{2}\\...\\x_{n}\end{bmatrix}}\,

{\vec  {x}}'={\begin{bmatrix}x_{2}\\x_{3}\\...\\g\end{bmatrix}}={\vec  {f}}(x,t)\,


Let {\vec  {f}}(x,t)\, be continuous for |t-t_{0}|\leq \alpha ,|{\vec  x}-{\vec  {x}}(0)|\leq \beta \, and satisfy and Lipschitz condition with constant L\, in this region. Let |{\vec  {f}}(x,t)|\leq M\, and \delta =\min \left\{\alpha ,{\frac  {\beta }{M}}\right\}\,. Then

{\vec  {x}}(t)={\vec  {x}}(0)+\int _{{t_{0}}}^{t}f({\vec  {x}}(s),s)ds\, has a unique solution for |t-t_{0}|\leq \delta \,.


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