Question

Express the 4th-order differential equation as a system of first order differential equations using matrix notation

Answer 1

\[y^{(4)} +\frac{ y^{(3)} }{t^2 }+\sin(t)y''+e^ty=t^2\]

Answer 2

i understand that i have to substitute the y derivatives as other variables, but i am really confused on how to convert it to a matrix

Answer 3

so far i have:
\[x_1=y, x_2=y', x_3=y'', x_4 = y^{(3)}\]

Answer 4

which gives me\[x_4'+\frac{ x_4 }{t^2 }+\sin(t)x_3 +e^tx_1 = t^2\]

so how do convert this using matrix notation?

Answer 5

Using the substitution you've made, you also have that
\[\begin{cases}
x_1=y\\
x_2=x_1'=y'\\
x_3=x_2'=y''\\
x_4=x_3'=y^{(3)}
\end{cases}\]
So you have the system of equations
\[\large\begin{cases}
x_1'=x_2\\\\
x_2'=x_3\\\\
x_3'=x_4\\\\
x_4'+\frac{1}{t^2}x_4+\sin t~x_3+e^t~x_1=t^2
\end{cases}\]

Answer 6

It shouldn't be too difficult to write that in matrix form.

Answer 7

okay thank you, i was confused that you had to make all the terms as derivative equations since your x'_4 is

Answer 8

You're welcome.