Question

Rewrite the given system as a first order system. $$x^{\prime \prime \prime }=f(t,x,y,y^{\prime })u^{\prime }=f(t,u,v,v^{\prime },w^{\prime })$$ (a) (b) $v^{\prime \prime }=g(t,u,v,v^{\prime },w)$ $$y^{\prime \prime }=g(t,y,y^{\prime })w^{\prime \prime }=h(t,u,v,v^{\prime },w,w^{\prime })$$ (c) $y^{\prime \prime \prime }=f(t,y,y^{\prime },y^{\prime \prime })$ (d) $y^{(4)}=f(t,y)$ $x^{\prime \prime }=f(t,x,y)$ (e) $y^{\prime \prime }=g(t,x,y)$

Short answer

Answer: The general approach to rewrite a system with higher-order derivatives as a first-order system involves introducing new variables for each derivative, starting with the first derivative and working up to the highest order derivative. Next, rewrite the given system using these new variables, expressing the higher-order derivatives in terms of the new variables. Finally, write out the new first-order system using the new variables and their relationships.

Step-by-step solution

(a) System with third order derivatives

Let's introduce the following variables to rewrite the system: $$\begin{gathered} u=x^{\prime } \\ {u}^{\prime }=x^{″} \\ v=x^{‴} \end{gathered}$$ Now, rewrite the third order derivative using the new variables: $$v=f(t,x,y,y^{\prime })$$ The first order system is now: $$\begin{gathered} u^{\prime }=u \\ {v}^{\prime }=x^{\prime } \\ v=f(t,x,y,y^{\prime }) \end{gathered}$$

(b) System with second order derivatives

Introduce new variables to rewrite the system: $$\begin{gathered} p=u^{\prime } \\ q=v^{\prime } \\ r=w^{\prime } \end{gathered}$$ Now, rewrite the given system using the new variables: $$\begin{gathered} u^{\prime }=p \\ {v}^{″}=g(t,u,v,p,w) \end{gathered}$$ The first order system is now: $$\begin{gathered} u^{\prime }=p \\ {v}^{\prime }=q \\ {q}^{\prime }=g(t,u,v,p,w) \\ {w}^{\prime }=r \end{gathered}$$

(c) System with third order derivatives

Introduce new variables to rewrite the system: $$\begin{gathered} p=y^{\prime } \\ q=y^{″} \\ r=y^{‴} \end{gathered}$$ Rewrite the given system using the new variables: $$y^{‴}=f(t,y,p,q)$$ The first order system is now: $$\begin{gathered} y^{\prime }=p \\ {y}^{″}=q \\ {y}^{‴}=r \\ r=f(t,y,p,q) \end{gathered}$$

(d) System with fourth order derivatives

Introduce new variables to rewrite the system: $$\begin{gathered} p=y^{\prime } \\ q=y^{″} \\ r=y^{‴} \\ s=y^{(4)} \end{gathered}$$ Rewrite the given system using the new variables: $$\begin{gathered} y^{(4)}=f(t,y) \\ {x}^{″}=f(t,x,y) \end{gathered}$$ The first order system is now: $$\begin{gathered} y^{\prime }=p \\ {y}^{″}=q \\ {y}^{‴}=r \\ {y}^{(4)}=s \\ s=f(t,y) \\ {x}^{\prime }=u \\ {x}^{″}=g(t,x,y) \end{gathered}$$

(e) System with second order derivatives

Introduce new variables to rewrite the system: $$\begin{gathered} p=x^{\prime } \\ q=y^{\prime } \end{gathered}$$ Rewrite the given system using the new variables: $$\begin{gathered} y^{″}=g(t,x,y) \\ {x}^{″}=f(t,x,y) \end{gathered}$$ The first order system is now: $$\begin{gathered} x^{\prime }=p \\ {x}^{″}=f(t,x,y) \\ {y}^{\prime }=q \\ {y}^{″}=g(t,x,y) \end{gathered}$$