Tutorial Week 6

Worksheet Questions

You can print the PDF. This worksheet relates to material from weeks 6 and 7.

1. Find the general solution to the following linear system of differential equations \begin{equation} \mathbf{x}'= \left( \begin{array}{cc} 2&-1\\ 1&4 \end{array} \right)\mathbf{x} \end{equation}




2. Consider the following system \begin{equation*} \mathbf{x}'= \left( \begin{array}{cc} 1&-2\\ 2&-4 \end{array} \right)\mathbf{x}+ \mathbf{b},\quad \text{where} \quad \mathbf{b}= \left( \begin{array}{c} b_1\\ b_2 \end{array} \right) \end{equation*}
1. Determine the general solution when $b_1=1$ and $b_2=2$.
2. In part (a), $b_1$ and $b_2$ were specially chosen in that the row reduced form of the matrix equation had a full row of zeros (including the RHS) and therefore a solution. Write down an equation for $b_1$ and $b_2$ that ensures this will happen.
3. Now consider the case with $b_1=3$ and $b_2=3$. Because this $\mathbf{b}$ does not satisfy the equation from part (b), a different form for your particular guess is needed. By analogy with second order systems, we guess $\mathbf{x_p} = t \mathbf{v}+\mathbf{w}$. Now we take the following steps to find out the general solution for this case:
   1. Plugging this $\mathbf{x_p} $ into the system of ODEs, we find that we must have $\mathbf{v} = t \mathbf{A} \mathbf{v}+ \mathbf{A} \mathbf{w} + \mathbf{b}$ for all $t$. This requires that $\mathbf{A} \mathbf{v}=0$ and $\mathbf{A} \mathbf{w}=\mathbf{v}-\mathbf{b};$
   2. $\mathbf{A} \mathbf{v}=0$ has a whole family of solutions. In fact , since $0$ is an eigenvalue of $\mathbf{A} $, $\mathbf{v}$ should be a corresponding eigenvector;
   3. Next consider the equation $\mathbf{A} \mathbf{w}=\mathbf{v}-\mathbf{b}$ which we must solve for $\mathbf{w}$. Notice that for any vector $\mathbf{w}$, $\mathbf{A} \mathbf{w}$ will always have a second component equal to twice its first component. Thus, to be able to solve $\mathbf{A} \mathbf{w}=\mathbf{v}-\mathbf{b}$ , we must make sure that $\mathbf{v}-\mathbf{b}$ has second component equal to twice its first component. Find the vector $\mathbf{v}$ from the family of solutions to $\mathbf{A} \mathbf{v}=0$ that does this. Then find $\mathbf{w}$ and write down the general solution for this case.

   
   

Tutorial Week 6 Solutions