Question

Rewrite the following differential equation as an equivalent system of first-order differential equations. Use the variables x1=x,x2=x′,x3=x′′,etc. (so that your equations will have the form x′2=x3, etc.: denote subscripts by appending the subscript to the variable name: x1= x1) Equation to rewrite: x(4) + 17x" - 17' + 17x = -9 cos(2t)

Answer 1

Answer:

See details below

Step-by-step explanation:

Your equation is $x^{(4)}+17x''-17x'+17x=-9\cos(2t)$. Solve for the fourth derivative to get $x^{(4)}=-17x''+17x'-17x-9\cos(2t)$

Now apply said change of variables: let $x_1=x,x_2=x',x_3=x'',x_4=x^{(3)}$. Then, substituting on out first equation, we obtain the system:

$x_4'=-17x_3+17x_2-17x_1-9\cos(2t)$

$x_4=x_3'$

$x_3=x_2'$

$x_2=x_1'$

In general, if you have an nth order ordinary differential equation, you can apply the same idea to obtain a system of differential equations with n unknowns. Therefore, solving systems of differential equations is equivalent to solving higher-order differential equations.