Problems

01

Writing $3^{\text{rd}}$ order linear ODE as $3$D system

Consider the ODE: $y^{\prime \prime \prime }-y^{\prime }+y=\cos t$. Rewrite this ODE as a $3$D system of equations, giving the matrix $\mathbf{A}$ and the inhomogeneous vector term $\mathbf{f}(t)$.

02

Converting system list to system matrix

Find the matrix + forcing vector formulations of the two ODE systems:

• (a) $$\begin{array}{rl}{x}^{\prime }(t)& =3x(t)-y(t)+t^2\\ y^{\prime }(t)& =-x(t)+2y(t)+e^t.\end{array}$$
• (b) $$\begin{array}{rl}{x}^{\prime }& =x+y+z\\ y^{\prime }& =2x-y+3z\\ z^{\prime }& =x+5z.\end{array}$$

03

Computing $3\times 3$ Wronskian

Find the Wronskian $W(t)$ for the following matrix:

$$\left(\begin{array}{rrr}{e}^t& e^{-t}& e^{2t}\\ e^t& -e^{-t}& 2e^{2t}\\ e^t& e^{-t}& 4e^{2t}\end{array}\right).$$

Are the column vectors of this matrix independent?

04

Derivative of a matrix function

Find the derivative of the following matrix-valued function:

$$\left(\begin{array}{rrr}\sin 2t& \cos 2t& e^{-2t}\\ -\sin 2t& 2\cos 2t& 3e^{-2t}\\ 3\sin 2t& \cos 2t& e^{-2t}\end{array}\right).$$

05

Verifying vector solution of ODE system

Show that the given vector function satisfies the given ODE system:

$$\mathbf{y}(t)=\left(\begin{array}{c}{e}^{3t}\\ 2e^{3t}\end{array}\right)\text{satisfies}{\mathbf{y}}^{\prime }=\left(\begin{array}{cc}1& 1\\ -2& 4\end{array}\right)\mathbf{y}.$$

06

Verifying fundamental matrix of ODE system

Show that the given matrix-valued function is a fundamental matrix for the given ODE system:

$$\mathbf{y}(t)=\left(\begin{array}{ccc}{e}^t& 0& 0\\ 0& e^t& e^{5t}\\ 0& e^t& -e^{5t}\end{array}\right)\text{satisfies}{\mathbf{y}}^{\prime }=\left(\begin{array}{rrr}1& 0& 0\\ 0& 3& -2\\ 0& -2& 3\end{array}\right)\mathbf{y}.$$