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$$\begin{array}{l}{y^{(6)}(t)=\left[y^{\prime}(t)\right]^{3}-\sin (y(t))+e^{2 t}} \\ {y(0)=y^{\prime}(0)=\cdots=y^{(5)}(0)=0}\end{array}$$

$x_{1}^{\prime}(t)=x_{2}(t)$

$x_{2}^{\prime}(t)=x_{3}(t)$

$x_{3}^{\prime}(t)=x_{4}(t)$

$x_{4}^{\prime}(t)=x_{5}(t)$

$x_{5}^{\prime}(t)=x_{6}(t)$

$x_{6}^{\prime}(t)=\left[x_{2}(t)\right]^{3}-\sin \left(x_{1}(t)\right)+e^{2 t}$

$x_{1}(0)=x_{2}(0)=$

$x_{3}(0)=x_{4}(0)=x_{5}(0)=x_{6}(0)=0$

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All right. So here we have the following differential equation. And now it is 1/6 order differential equation. So the six derivative of why is the first derivative of why cubed minus sign of why plus u to the T. Okay. And here why, of course, is a function of tea. And we have some initial conditions in the initial conditions. Are that why? So the ice derivative of why evaluated at zero is equal to zero, and here I'll just making notes. So eyes between zero and five. And when I say why of zero. So the zeroth derivative, Why, I just simply mean why, Okay, Sounds good to know. What we want to do is convert this to a system of first order equations. So now this is 1/6 order equation. So what that means is that we can convert this into a system of six first order questions. So a six or sixth order equation converts to six first word or questions. And what we do is we define the following variable. So x one is just going to be why, okay, it's about to start with why x two will just be the derivative said why? Crime X three will be Why double prime and then go all the way down The final variable x six will be the fifth driven of Why so ex of I will always just be the eye minus one to rivet of life. Okay. And we have six variables because it's 1/6 order different show equation. And now if we look att x six and take the derivative, that's just the sixth derivative of why? Because the derivative of the fifth derivative is the sixth murder. And now we have why Pop Prime Cube. But why? Prime is X two. So this is X two cute minus sign of why, But why is X one plus even the d. So there's one of our equations. Can we have five more? So he had X five. If I take the derivative of X five and I remember ex spies is the fourth derivative of wind. That's going to give me the fifth derivative of why which is X six and the next four prime is gonna be x five X speed prime is gonna be X for in the annex to crime. It's gonna be x three and then X one Prime is going to be extreme. Okay, So here's our six equations. 123456 And then our initial conditions for this system of six equations is just going to be ex of I of zero is equal to zero. And here I is gonna be between one in six. So here is our equation. And Standard form has assistant the first order equations.

Georgia Southern University

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