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 \((a,b)\).