# y &#x2034; </msup> ( t ) + s i n ( t ) y &#x2033; </m

Cristopher Knox 2022-06-30 Answered
${y}^{‴}\left(t\right)+sin\left(t\right){y}^{″}\left(t\right)-g\left(t\right){y}^{\prime }\left(t\right)+g\left(t\right)y\left(t\right)=f\left(t\right)$
Write the following third order differential equation as an equivalent system of first order ordinary differential equations and write it in the form
$\frac{dz}{dt}=f\left(t,z\right)$
Here is what I have so far:
${x}_{1}\left(t\right)=y\left(t\right)$
${x}_{2}\left(t\right)={y}^{\prime }\left(t\right)$
${x}_{3}\left(t\right)={y}^{″}\left(t\right)$
so I can then do,
${x}_{1}^{\prime }\left(t\right)={y}^{\prime }\left(t\right)$
${x}_{2}^{\prime }\left(t\right)={y}^{″}\left(t\right)$
${x}_{3}^{\prime }\left(t\right)={y}^{‴}\left(t\right)=f\left(t\right)-sin\left(t\right){y}^{″}\left(t\right)+g\left(t\right){y}^{\prime }\left(t\right)-g\left(t\right)y\left(t\right)$
${x}_{3}^{\prime }\left(t\right)=f\left(t\right)-sin\left(t\right){x}_{3}\left(t\right)+g\left(t\right){x}_{2}\left(t\right)-g\left(t\right){x}_{1}\left(t\right)$
and I think the dynamics function is:
$f\left(t,z\right)=\left[{x}_{1}^{\prime }\left(t\right),{x}_{2}^{″}\left(t\right),{x}_{3}^{‴}\left(t\right)\right]$
$f\left(t,z\right)=\left[{x}_{2}\left(t\right),{x}_{3}\left(t\right),f\left(t\right)-sin\left(t\right){x}_{3}\left(t\right)+g\left(t\right){x}_{2}\left(t\right)-g\left(t\right){x}_{1}\left(t\right)\right]$
I'm not sure if I'm doing this correctly. What does difference does the f(t) and g(t) make in this case? what is z exactly?
You can still ask an expert for help

• Live experts 24/7
• Questions are typically answered in as fast as 30 minutes
• Personalized clear answers

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Nathen Austin
Remember that your final goal is to obtain a system of FIRST order equations. So, any higher derivatives need to be rewritten appropriately.
Using your notation, we have
$\begin{array}{rcl}{x}_{1}^{\prime }& =& {x}_{2}\\ {x}_{2}^{\prime }& =& {x}_{3}\\ {x}_{3}^{\prime }& =& f\left(t\right)-\mathrm{sin}\left(t\right){x}_{3}+g\left(t\right){x}_{2}-g\left(t\right){x}_{1}\end{array}$
Here z would represent the 3-vector $\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right)$
Try and think of the entire set of expressions on the right hand side as the output of multivariable function from ${\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$. (It's not the best idea to use f again in this context. Think of f(t,z) as $f\left(t,{x}_{1},{x}_{2},{x}_{3}\right)$)