10.2E: Linear Systems of Differential Equations (Exercises)

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Q10.2.1

1. Rewrite the system in matrix form and verify that the given vector function satisfies the system for any choice of the constants $c_1$ and $c_2$.

1. $\begin{array}{ccl}y'_1&=&2y_1 + 4y_2\\[4pt] y_2'&=&4y_1+2y_2;\end{array} \quad {\bf y}=c_1\twocol11e^{6t}+c_2\twocol1{-1}e^{-2t}$
2. $\begin{array}{ccl}y'_1&=&-2y_1 - 2y_2\\[4pt] y_2'&=&-5y_1 + \phantom{2}y_2;\end{array} \quad {\bf y}=c_1\twocol11e^{-4t}+c_2\twocol{-2}5e^{3t}$
3. $\begin{array}{ccr}y'_1&=&-4y_1 -10y_2\\[4pt] y_2'&=&3y_1 + \phantom{1}7y_2;\end{array} \quad {\bf y}=c_1\twocol{-5}3e^{2t}+c_2\twocol2{-1}e^t$
4. $\begin{array}{ccl}y'_1&=&2y_1 +\phantom{2}y_2 \\[4pt] y_2'&=&\phantom{2}y_1 + 2y_2;\end{array} \quad {\bf y}=c_1\twocol11e^{3t}+c_2\twocol1{-1}e^t$

2. Rewrite the system in matrix form and verify that the given vector function satisfies the system for any choice of the constants $c_1$, $c_2$, and $c_3$.

1. $\begin{array}{ccr}y'_1&=&- y_1+2y_2 + 3y_3 \\[4pt] y_2'&=&y_2 + 6y_3\\[4pt]y_3'&=&- 2y_3;\end{array}$
   ${\bf y}=c_1\threecol110e^t+c_2\threecol100e^{-t}+c_3\threecol1{-2}1e^{-2t}$
2. $\begin{array}{ccc}y'_1&=&\phantom{2y_1+}2y_2 + 2y_3 \\[4pt] y_2'&=&2y_1\phantom{+2y_2} + 2y_3\\[4pt]y_3'&=&2y_1 + 2y_2;\phantom{+2y_3}\end{array}$
   ${\bf y}=c_1\threecol{-1}01e^{-2t}+c_2\threecol0{-1}1e^{-2t}+c_3\threecol111e^{4t}$
3. $\begin{array}{ccr}y'_1&=&-y_1 +2y_2 + 2y_3\\[4pt] y_2'&=&2y_1 -\phantom{2}y_2 +2y_3\\[4pt]y_3'&=&2y_1 + 2y_2 -\phantom{2}y_3;\end{array}$
   ${\bf y}=c_1\threecol{-1}01e^{-3t}+c_2\threecol0{-1}1e^{-3t}+c_3\threecol111e^{3t}$
4. $\begin{array}{ccr}y'_1&=&3y_1 - \phantom{2}y_2 -\phantom{2}y_3 \\[4pt] y_2'&=&-2y_1 + 3y_2 + 2y_3\\[4pt]y_3'&=&\phantom{-}4y_1 -\phantom{3}y_2 - 2y_3;\end{array}$
   ${\bf y}=c_1\threecol101e^{2t}+c_2\threecol1{-1}1e^{3t}+c_3\threecol1{-3}7e^{-t}$

3. Rewrite the initial value problem in matrix form and verify that the given vector function is a solution.

1. $\begin{array}{ccl}y'_1 &=&\phantom{-2}y_1+\phantom{4}y_2\\[4pt] y_2'&=&-2y_1 + 4y_2,\end{array} \begin{array}{ccr}y_1(0)&=&1\\[4pt]y_2(0)&=&0;\end{array}$ ${\bf y}=2\twocol11e^{2t}-\twocol12e^{3t}$
2. $\begin{array}{ccl}y'_1 &=&5y_1 + 3y_2 \\[4pt] y_2'&=&- y_1 + y_2,\end{array} \begin{array}{ccr}y_1(0)&=&12\\[4pt]y_2(0)&=&-6;\end{array}$ ${\bf y}=3\twocol1{-1}e^{2t}+3\twocol3{-1}e^{4t}$

4. Rewrite the initial value problem in matrix form and verify that the given vector function is a solution.

1. $\begin{array}{ccr}y'_1&=&6y_1 + 4y_2 + 4y_3 \\[4pt] y_2'&=&-7y_1 -2y_2 - y_3,\\[4pt]y_3'&=&7y_1 + 4y_2 + 3y_3\end{array},\; \begin{array}{ccr}y_1(0)&=&3\\[4pt] y_2(0)&=&-6\\[4pt] y_3(0)&=&4\end{array}$
   ${\bf y}=\threecol1{-1}1e^{6t}+2\threecol1{-2}1e^{2t}+\threecol0{-1}1e^{-t}$
2. $\begin{array}{ccr}y'_1&=& \phantom{-}8y_1 + 7y_2 +\phantom{1}7y_3 \\[4pt] y_2'&=&-5y_1 -6y_2 -\phantom{1}9y_3,\\[4pt]y_3'&=& \phantom{-}5y_1 + 7y_2 +10y_3,\end{array}\ \begin{array}{ccr}y_1(0)&=&2\\[4pt] y_2(0)&=&-4\\[4pt] y_3(0)&=&3\end{array}$
   ${\bf y}=\threecol1{-1}1e^{8t}+\threecol0{-1}1e^{3t}+\threecol1{-2}1e^t$

5. Rewrite the system in matrix form and verify that the given vector function satisfies the system for any choice of the constants $c_1$ and $c_2$.

1. $\begin{array}{ccc}y'_1&=&-3y_1+2y_2+3-2t \\[4pt] y_2'&=&-5y_1+3y_2+6-3t\end{array}$
   ${\bf y}=c_1\left[\begin{array}{c}2\cos t\\[4pt]3\cos t-\sin t\end{array}\right]+c_2\left[\begin{array}{c}2\sin t\\[4pt]3\sin t+\cos t \end{array}\right]+\twocol1t$
2. $\begin{array}{ccc}y'_1&=&3y_1+y_2-5e^t \\[4pt] y_2'&=&-y_1+y_2+e^t\end{array}$
   ${\bf y}=c_1\twocol{-1}1e^{2t}+c_2\left[\begin{array}{c}1+t\\[4pt]-t\end{array} \right]e^{2t}+\twocol13e^t$
3. $\begin{array}{ccl}y'_1&=&-y_1-4y_2+4e^t+8te^t \\[4pt] y_2'&=&-y_1-\phantom{4}y_2+e^{3t}+(4t+2)e^t\end{array}$
   ${\bf y}=c_1\twocol21e^{-3t}+c_2\twocol{-2}1e^t+\left[\begin{array}{c} e^{3t}\\[4pt]2te^t\end{array}\right]$
4. $\begin{array}{ccc}y'_1&=&-6y_1-3y_2+14e^{2t}+12e^t \\[4pt] y_2'&=&\phantom{6}y_1-2y_2+7e^{2t}-12e^t\end{array}$
   ${\bf y}=c_1\twocol{-3}1e^{-5t}+c_2\twocol{-1}1e^{-3t}+ \left[\begin{array}{c}e^{2t}+3e^t\\[4pt]2e^{2t}-3e^t\end{array}\right]$

6. Convert the linear scalar equation

$$P_0(t)y^{(n)}+P_1(t)y^{(n-1)}+\cdots+P_n(t)y(t)=F(t) \tag{A}$$

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

$$Q(t)=\left[\begin{array}{cccc}{q_{11}(t)}&{q_{12}(t)}&{\cdots }&{q_{1s}(t)} \\[4pt] {q_{21}(t)}&{q_{22}(t)}&{\cdots }&{q_{2s}(t)} \\[4pt] {\vdots }&{\vdots }&{\ddots }&{\vdots } \\[4pt] {q_{r1}(t)}&{q_{r2}(t)}&{\cdots }&{q_{rs}(t)} \end{array} \right] \nonumber$$

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

$$Q(t)=\left[\begin{array}{cccc}{q'_{11}(t)}&{q'_{12}(t)}&{\cdots }&{q'_{1s}(t)} \\[4pt] {q'_{21}(t)}&{q'_{22}(t)}&{\cdots }&{q'_{2s}(t)} \\[4pt] {\vdots }&{\vdots }&{\ddots }&{\vdots } \\[4pt] {q'_{r1}(t)}&{q'_{r2}(t)}&{\cdots }&{q'_{rs}(t)} \end{array} \right] \nonumber$$

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

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

$$Y= \left[\begin{array}{cc}{y_{11}}&{y_{12}}\\[4pt]{y_{21}}&{y_{22}}\end{array}\right].\nonumber$$

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 $(a,b)$.