10.2.1: 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 \(c_2\).
- \(\begin{array}{ccl}y'_1&=&2y_1 + 4y_2\\ y_2'&=&4y_1+2y_2;\end{array} \quad {\bf y}=c_1\twocol11e^{6t}+c_2\twocol1{-1}e^{-2t}\)
- \(\begin{array}{ccl}y'_1&=&-2y_1 - 2y_2\\ y_2'&=&-5y_1 + \phantom{2}y_2;\end{array} \quad {\bf y}=c_1\twocol11e^{-4t}+c_2\twocol{-2}5e^{3t}\)
- \(\begin{array}{ccr}y'_1&=&-4y_1 -10y_2\\ y_2'&=&3y_1 + \phantom{1}7y_2;\end{array} \quad {\bf y}=c_1\twocol{-5}3e^{2t}+c_2\twocol2{-1}e^t\)
- \(\begin{array}{ccl}y'_1&=&2y_1 +\phantom{2}y_2 \\ y_2'&=&\phantom{2}y_1 + 2y_2;\end{array} \quad {\bf y}=c_1\twocol11e^{3t}+c_2\twocol1{-1}e^t\)
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 \(c_3\).
- \(\begin{array}{ccr}y'_1&=&- y_1+2y_2 + 3y_3 \\ y_2'&=&y_2 + 6y_3\\y_3'&=&- 2y_3;\end{array}\)
\({\bf y}=c_1\threecol110e^t+c_2\threecol100e^{-t}+c_3\threecol1{-2}1e^{-2t}\) - \(\begin{array}{ccc}y'_1&=&\phantom{2y_1+}2y_2 + 2y_3 \\ y_2'&=&2y_1\phantom{+2y_2} + 2y_3\\y_3'&=&2y_1 + 2y_2;\phantom{+2y_3}\end{array}\)
\({\bf y}=c_1\threecol{-1}01e^{-2t}+c_2\threecol0{-1}1e^{-2t}+c_3\threecol111e^{4t}\) - \(\begin{array}{ccr}y'_1&=&-y_1 +2y_2 + 2y_3\\ y_2'&=&2y_1 -\phantom{2}y_2 +2y_3\\y_3'&=&2y_1 + 2y_2 -\phantom{2}y_3;\end{array}\)
\({\bf y}=c_1\threecol{-1}01e^{-3t}+c_2\threecol0{-1}1e^{-3t}+c_3\threecol111e^{3t}\) - \(\begin{array}{ccr}y'_1&=&3y_1 - \phantom{2}y_2 -\phantom{2}y_3 \\ y_2'&=&-2y_1 + 3y_2 + 2y_3\\y_3'&=&\phantom{-}4y_1 -\phantom{3}y_2 - 2y_3;\end{array}\)
\({\bf y}=c_1\threecol101e^{2t}+c_2\threecol1{-1}1e^{3t}+c_3\threecol1{-3}7e^{-t}\)
3. Rewrite the initial value problem in matrix form and verify that the given vector function is a solution.
- \(\begin{array}{ccl}y'_1 &=&\phantom{-2}y_1+\phantom{4}y_2\\ y_2'&=&-2y_1 + 4y_2,\end{array} \begin{array}{ccr}y_1(0)&=&1\\y_2(0)&=&0;\end{array}\) \({\bf y}=2\twocol11e^{2t}-\twocol12e^{3t}\)
- \(\begin{array}{ccl}y'_1 &=&5y_1 + 3y_2 \\ y_2'&=&- y_1 + y_2,\end{array} \begin{array}{ccr}y_1(0)&=&12\\y_2(0)&=&-6;\end{array}\) \({\bf y}=3\twocol1{-1}e^{2t}+3\twocol3{-1}e^{4t}\)
4. Rewrite the initial value problem in matrix form and verify that the given vector function is a solution.
- \(\begin{array}{ccr}y'_1&=&6y_1 + 4y_2 + 4y_3 \\ y_2'&=&-7y_1 -2y_2 - y_3,\\y_3'&=&7y_1 + 4y_2 + 3y_3\end{array},\; \begin{array}{ccr}y_1(0)&=&3\\ y_2(0)&=&-6\\ y_3(0)&=&4\end{array}\)
\({\bf y}=\threecol1{-1}1e^{6t}+2\threecol1{-2}1e^{2t}+\threecol0{-1}1e^{-t}\) - \(\begin{array}{ccr}y'_1&=& \phantom{-}8y_1 + 7y_2 +\phantom{1}7y_3 \\ y_2'&=&-5y_1 -6y_2 -\phantom{1}9y_3,\\y_3'&=& \phantom{-}5y_1 + 7y_2 +10y_3,\end{array}\ \begin{array}{ccr}y_1(0)&=&2\\ y_2(0)&=&-4\\ y_3(0)&=&3\end{array}\)
\({\bf y}=\threecol1{-1}1e^{8t}+\threecol0{-1}1e^{3t}+\threecol1{-2}1e^t\)
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 \(c_2\).
- \(\begin{array}{ccc}y'_1&=&-3y_1+2y_2+3-2t \\ y_2'&=&-5y_1+3y_2+6-3t\end{array}\)
\({\bf y}=c_1\left[\begin{array}{c}2\cos t\\3\cos t-\sin t\end{array}\right]+c_2\left[\begin{array}{c}2\sin t\\3\sin t+\cos t \end{array}\right]+\twocol1t\) - \(\begin{array}{ccc}y'_1&=&3y_1+y_2-5e^t \\ y_2'&=&-y_1+y_2+e^t\end{array}\)
\({\bf y}=c_1\twocol{-1}1e^{2t}+c_2\left[\begin{array}{c}1+t\\-t\end{array} \right]e^{2t}+\twocol13e^t\) - \(\begin{array}{ccl}y'_1&=&-y_1-4y_2+4e^t+8te^t \\ y_2'&=&-y_1-\phantom{4}y_2+e^{3t}+(4t+2)e^t\end{array}\)
\({\bf y}=c_1\twocol21e^{-3t}+c_2\twocol{-2}1e^t+\left[\begin{array}{c} e^{3t}\\2te^t\end{array}\right]\) - \(\begin{array}{ccc}y'_1&=&-6y_1-3y_2+14e^{2t}+12e^t \\ y_2'&=&\phantom{6}y_1-2y_2+7e^{2t}-12e^t\end{array}\)
\({\bf y}=c_1\twocol{-3}1e^{-5t}+c_2\twocol{-1}1e^{-3t}+ \left[\begin{array}{c}e^{2t}+3e^t\\2e^{2t}-3e^t\end{array}\right]\)
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
\[Q(t)=\left[\begin{array}{cccc}{q_{11}(t)}&{q_{12}(t)}&{\cdots }&{q_{1s}(t)} \\ {q_{21}(t)}&{q_{22}(t)}&{\cdots }&{q_{2s}(t)} \\ {\vdots }&{\vdots }&{\ddots }&{\vdots } \\ {q_{r1}(t)}&{q_{r2}(t)}&{\cdots }&{q_{rs}(t)} \end{array} \right] \nonumber\]
is said to be differentiable if its entries \(\{q_{ij}\}\) are differentiable. Then the derivative \(Q'\) is defined by
\[Q(t)=\left[\begin{array}{cccc}{q'_{11}(t)}&{q'_{12}(t)}&{\cdots }&{q'_{1s}(t)} \\ {q'_{21}(t)}&{q'_{22}(t)}&{\cdots }&{q'_{2s}(t)} \\ {\vdots }&{\vdots }&{\ddots }&{\vdots } \\ {q'_{r1}(t)}&{q'_{r2}(t)}&{\cdots }&{q'_{rs}(t)} \end{array} \right] \nonumber\]
- Prove: If \(P\) and \(Q\) are differentiable matrices such that \(P+Q\) is defined and if \(c_1\) and \(c_2\) are constants, then \[(c_1P+c_2Q)'=c_1P'+c_2Q'.\nonumber \]
- Prove: If \(P\) and \(Q\) are differentiable matrices such that \(PQ\) is defined, then \[(PQ)'=P'Q+PQ'.\nonumber \]
8. Verify that \(Y' = AY\).
- \(Y=\left[\begin{array}{cc}{e^{6t}}&{e^{-2t}}\\{e^{6t}}&{-e^{-2t}} \end{array} \right],\quad A=\left[\begin{array}{cc}{2}&{4}\\{4}&{2} \end{array} \right]\)
- \(Y=\left[\begin{array}{cc}{e^{-4t}}&{-2e^{3t}}\\{e^{-4t}}&{5e^{3t}} \end{array} \right],\quad A=\left[\begin{array}{cc}{-2}&{-2}\\{-5}&{1} \end{array} \right]\)
- \(Y=\left[\begin{array}{cc}{-5e^{2t}}&{2e^{t}}\\{3e^{2t}}&{-e^{t}} \end{array} \right],\quad A=\left[\begin{array}{cc}{-4}&{-10}\\{3}&{7} \end{array} \right]\)
- \(Y=\left[\begin{array}{cc}{e^{3t}}&{e^{t}}\\{e^{3t}}&{-e^{t}} \end{array} \right],\quad A=\left[\begin{array}{cc}{2}&{1}\\{1}&{2} \end{array} \right]\)
- \(Y = \left[\begin{array}{ccc} e^t&e^{-t}& e^{-2t}\\ e^t&0&-2e^{-2t}\\ 0&0&e^{-2t}\end{array}\right], \quad A = \left[\begin{array}{ccc}{-1}&{2}&{3}\\{0}&{1}&{6}\\{0}&{0}&{-2} \end{array} \right]\)
- \(Y = \left[\begin{array}{ccc} {-e^{-2t}}&{-e^{-2t}}& {e^{4t}}\\ {0}&{e^{-2t}}&{e^{4t}}\\ {e^{-2t}}&{0}&{e^{4t}}\end{array}\right], \quad A = \left[\begin{array}{ccc}{0}&{2}&{2}\\{2}&{0}&{2}\\{2}&{2}&{0} \end{array} \right]\)
- \(Y = \left[\begin{array}{ccc} {e^{3t}}&{e^{-3t}}& {0}\\ {e^{3t}}&{0}&{-e^{-3t}}\\ {e^{3t}}&{e^{-3t}}&{e^{-3t}}\end{array}\right], \quad A = \left[\begin{array}{ccc}{-9}&{6}&{6}\\{-6}&{3}&{6}\\{-6}&{6}&{3} \end{array} \right]\)
- \(Y = \left[\begin{array}{ccc} {e^{2t}}&{e^{3t}}& {e^{-t}}\\ {0}&{-e^{3t}}&{-3e^{-t}}\\ {e^{2t}}&{e^{3t}}&{7e^{-t}}\end{array}\right], \quad A = \left[\begin{array}{ccc}{3}&{-1}&{-1}\\{-2}&{3}&{2}\\{4}&{-1}&{-2} \end{array} \right]\)
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
\[Y= \left[\begin{array}{cc}{y_{11}}&{y_{12}}\\{y_{21}}&{y_{22}}\end{array}\right].\nonumber \]
- Show that \(Y'=AY\).
- Show that if \({\bf c}\) is a constant vector then \({\bf y}= Y{\bf c}\) is a solution of (A).
- State generalizations of (a) and (b) for \(n\times n\) systems.
10. Suppose \(Y\) is a differentiable square matrix.
- Find a formula for the derivative of \(Y^2\).
- Find a formula for the derivative of \(Y^n\), where \(n\) is any positive integer.
- 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.
- Show that (\(Y^{-1})'= -Y^{-1}Y'Y^{-1}\). (Hint: Differentiate the identity \(Y^{-1}Y=I\).)
- Find the derivative of \(Y^{-n}=\left(Y^{-1}\right)^n\), where \(n\) is a positive integer.
- 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 \((a,b)\).