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# 10.2.1: Linear Systems of Differential Equations (Exercises)



into an equivalent $$n\times n$$ system

${\bf y'}=A(t){\bf y}+{\bf f}(t),\nonumber$

and show that $$A$$ and $${\bf f}$$ are continuous on an interval $$(a,b)$$ if and only if (A) is normal on $$(a,b)$$.

7. A matrix function

$Q(t)=\left[\begin{array}{cccc}{q_{11}(t)}&{q_{12}(t)}&{\cdots }&{q_{1s}(t)} \\ {q_{21}(t)}&{q_{22}(t)}&{\cdots }&{q_{2s}(t)} \\ {\vdots }&{\vdots }&{\ddots }&{\vdots } \\ {q_{r1}(t)}&{q_{r2}(t)}&{\cdots }&{q_{rs}(t)} \end{array} \right] \nonumber$

is said to be differentiable if its entries $$\{q_{ij}\}$$ are differentiable. Then the derivative $$Q'$$ is defined by

$Q(t)=\left[\begin{array}{cccc}{q'_{11}(t)}&{q'_{12}(t)}&{\cdots }&{q'_{1s}(t)} \\ {q'_{21}(t)}&{q'_{22}(t)}&{\cdots }&{q'_{2s}(t)} \\ {\vdots }&{\vdots }&{\ddots }&{\vdots } \\ {q'_{r1}(t)}&{q'_{r2}(t)}&{\cdots }&{q'_{rs}(t)} \end{array} \right] \nonumber$

1. Prove: If $$P$$ and $$Q$$ are differentiable matrices such that $$P+Q$$ is defined and if $$c_1$$ and $$c_2$$ are constants, then $(c_1P+c_2Q)'=c_1P'+c_2Q'.\nonumber$
2. Prove: If $$P$$ and $$Q$$ are differentiable matrices such that $$PQ$$ is defined, then $(PQ)'=P'Q+PQ'.\nonumber$

8. Verify that $$Y' = AY$$.

1. $$Y=\left[\begin{array}{cc}{e^{6t}}&{e^{-2t}}\\{e^{6t}}&{-e^{-2t}} \end{array} \right],\quad A=\left[\begin{array}{cc}{2}&{4}\\{4}&{2} \end{array} \right]$$
2. $$Y=\left[\begin{array}{cc}{e^{-4t}}&{-2e^{3t}}\\{e^{-4t}}&{5e^{3t}} \end{array} \right],\quad A=\left[\begin{array}{cc}{-2}&{-2}\\{-5}&{1} \end{array} \right]$$
3. $$Y=\left[\begin{array}{cc}{-5e^{2t}}&{2e^{t}}\\{3e^{2t}}&{-e^{t}} \end{array} \right],\quad A=\left[\begin{array}{cc}{-4}&{-10}\\{3}&{7} \end{array} \right]$$
4. $$Y=\left[\begin{array}{cc}{e^{3t}}&{e^{t}}\\{e^{3t}}&{-e^{t}} \end{array} \right],\quad A=\left[\begin{array}{cc}{2}&{1}\\{1}&{2} \end{array} \right]$$
5. $$Y = \left[\begin{array}{ccc} e^t&e^{-t}& e^{-2t}\\ e^t&0&-2e^{-2t}\\ 0&0&e^{-2t}\end{array}\right], \quad A = \left[\begin{array}{ccc}{-1}&{2}&{3}\\{0}&{1}&{6}\\{0}&{0}&{-2} \end{array} \right]$$
6. $$Y = \left[\begin{array}{ccc} {-e^{-2t}}&{-e^{-2t}}& {e^{4t}}\\ {0}&{e^{-2t}}&{e^{4t}}\\ {e^{-2t}}&{0}&{e^{4t}}\end{array}\right], \quad A = \left[\begin{array}{ccc}{0}&{2}&{2}\\{2}&{0}&{2}\\{2}&{2}&{0} \end{array} \right]$$
7. $$Y = \left[\begin{array}{ccc} {e^{3t}}&{e^{-3t}}& {0}\\ {e^{3t}}&{0}&{-e^{-3t}}\\ {e^{3t}}&{e^{-3t}}&{e^{-3t}}\end{array}\right], \quad A = \left[\begin{array}{ccc}{-9}&{6}&{6}\\{-6}&{3}&{6}\\{-6}&{6}&{3} \end{array} \right]$$
8. $$Y = \left[\begin{array}{ccc} {e^{2t}}&{e^{3t}}& {e^{-t}}\\ {0}&{-e^{3t}}&{-3e^{-t}}\\ {e^{2t}}&{e^{3t}}&{7e^{-t}}\end{array}\right], \quad A = \left[\begin{array}{ccc}{3}&{-1}&{-1}\\{-2}&{3}&{2}\\{4}&{-1}&{-2} \end{array} \right]$$

9. Suppose

${\bf y}_1=\twocol{y_{11}}{y_{21}}\quad \text{and} \quad{\bf y}_2=\twocol{y_{12}}{y_{22}}\nonumber$

are solutions of the homogeneous system

${\bf y}'=A(t){\bf y}, \tag{A}$

and define

$Y= \left[\begin{array}{cc}{y_{11}}&{y_{12}}\\{y_{21}}&{y_{22}}\end{array}\right].\nonumber$

1. Show that $$Y'=AY$$.
2. Show that if $${\bf c}$$ is a constant vector then $${\bf y}= Y{\bf c}$$ is a solution of (A).
3. State generalizations of (a) and (b) for $$n\times n$$ systems.

10. Suppose $$Y$$ is a differentiable square matrix.

1. Find a formula for the derivative of $$Y^2$$.
2. Find a formula for the derivative of $$Y^n$$, where $$n$$ is any positive integer.
3. State how the results obtained in (a) and (b) are analogous to results from calculus concerning scalar functions.

11. It can be shown that if $$Y$$ is a differentiable and invertible square matrix function, then $$Y^{-1}$$ is differentiable.

1. Show that ($$Y^{-1})'= -Y^{-1}Y'Y^{-1}$$. (Hint: Differentiate the identity $$Y^{-1}Y=I$$.)
2. Find the derivative of $$Y^{-n}=\left(Y^{-1}\right)^n$$, where $$n$$ is a positive integer.
3. State how the results obtained in (a) and (b) are analogous to results from calculus concerning scalar functions.

12. Show that Theorem 10.2.1 implies Theorem 9.1.1. HINT: Write the scalar function $P_{0}(x)y^{(n)}+P_{1}(x)y^{(n-1)}+\cdots +P_{n}(x)y=F(x)\nonumber$ as an $$n\times n$$ system of linear equations.

13. Suppose $${\bf y}$$ is a solution of the $$n\times n$$ system $${\bf y}'=A(t){\bf y}$$ on $$(a,b)$$, and that the $$n\times n$$ matrix $$P$$ is invertible and differentiable on $$(a,b)$$. Find a matrix $$B$$ such that the function $${\bf x}=P{\bf y}$$ is a solution of $${\bf x}'=B{\bf x}$$ on $$(a,b)$$.

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