Chapter 9

Verify that \(\mathbf{x}_{1}(t)=\left[\begin{array}{l}e^{t^{2}-t} \\\ -1\end{array}\right]\) and \(\mathbf{x}_{2}(t)=\left[\begin{array}{c}0 \\ 2 e^{t}\end{array}\right]\) are linearly independent solutions to \(\mathbf{x}^{\prime}=A \mathbf{x},\) where $$ A=\left[\begin{array}{ll} 2 t-1 & 0 \\ e^{t-t^{2}} & 1 \end{array}\right] $$ Write the general solution to the system \(\mathbf{x}^{\prime}=A \mathbf{x}\)

Determine the general solution to the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\) $$\left[\begin{array}{rr} 1 & 1 \\ -1 & 3 \end{array}\right]$$

Determine all equilibrium points of the given system and, if possible, characterize them as centers, spirals, saddles, or nodes. $$x^{\prime}=y(3 x-2), \quad y^{\prime}=2 x+9 y^{2}$$

If \(X(t)\) is any fundamental matrix for \(\mathbf{x}^{\prime}=A \mathbf{x},\) show that the transition matrix based at \(t=0\) is given by $$X_{0}=X(t) X^{-1}(0)$$.

Use the variation-of-parameters technique to find a particular solution \(\mathbf{x}_{p}\) to \(\mathbf{x}^{\prime}=A \mathbf{x}+\mathbf{b},\) for the given \(A\) and \(\mathbf{b} .\) Also obtain the general solution to the system of differential equations. $$A=\left[\begin{array}{ll} 4 & -3 \\ 2 & -1 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l} e^{2 t} \\ e^{t} \end{array}\right]$$

Show that the given functions are solutions of the system \(\mathbf{x}^{\prime}(t)=A(x) \mathbf{x}(t)\) for the given matrix \(A,\) and hence, find the general solution to the system (remember to check linear independence). If auxiliary conditions are given, find the particular solution that satisfies these conditions. $$\begin{aligned} &\mathbf{x}_{1}(t)=\left[\begin{array}{c} \sin 3 t \\ \cos 3 t \end{array}\right], \quad \mathbf{x}_{2}(t)=\left[\begin{array}{r} -\cos 3 t \\ \sin 3 t \end{array}\right],\\\ &A=\left[\begin{array}{rr} 0 & 3 \\ -3 & 0 \end{array}\right]. \end{aligned}$$

Show that the given vector functions are linearly independent on \((-\infty, \infty)\). $$\mathbf{x}_{1}(t)=\left[\begin{array}{r} e^{t} \\ -e^{t} \end{array}\right], \quad \mathbf{x}_{2}(t)=\left[\begin{array}{l} e^{t} \\ e^{t} \end{array}\right]$$

Solve the given system of differential equations. $$x_{1}^{\prime}=2 x_{1}+x_{2}, \quad x_{2}^{\prime}=2 x_{1}+3 x_{2}$$

Show that the given functions are solutions of the system \(\mathbf{x}^{\prime}(t)=A(x) \mathbf{x}(t)\) for the given matrix \(A,\) and hence, find the general solution to the system (remember to check linear independence). If auxiliary conditions are given, find the particular solution that satisfies these conditions. $$\begin{aligned} &\mathbf{x}_{1}(t)=\left[\begin{array}{c} e^{4 t} \\ 2 e^{4 t} \end{array}\right], \quad \mathbf{x}_{2}(t)=\left[\begin{array}{c} 3 e^{-t} \\ e^{-t} \end{array}\right],\\\ &A=\left[\begin{array}{ll} -2 & 3 \\\ -2 & 5 \end{array}\right], \quad \mathbf{x}(0)=\left[\begin{array}{r} -2 \\\ 1 \end{array}\right]. \end{aligned}$$

Consider the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}+\mathbf{b},\) where $$ \begin{array}{l} A=\left[\begin{array}{ccc} t \cot \left(t^{2}\right) & 0 & t \cos \left(t^{2}\right) / 2 \\ 0 & 1 / t & -1 \\ \csc \left(t^{2}\right) & 1 & -1 \end{array}\right] \\ \mathbf{b}=\left[\begin{array}{c} 0 \\ 2-t \sin t \\ 1-t \cos t \end{array}\right] \end{array} $$ (a) Verify that \(\mathbf{x}(t)=\left[\begin{array}{c}\sin \left(t^{2}\right) \\\ t \cos t \\ 2\end{array}\right]\) is a solution to this system. (b) Is it possible for a constant vector \(\mathbf{x}_{0}\) to solve the system? Justify your answer.

Determine the general solution to the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\) $$\left[\begin{array}{rr} 0 & -2 \\ 2 & 4 \end{array}\right]$$

Determine all equilibrium points of the given system and, if possible, characterize them as centers, spirals, saddles, or nodes. $$x^{\prime}=y(3 x-2), \quad y^{\prime}=2 x-9 y^{2}$$

Use the techniques from Section 9.4 and Section 9.5 to determine a fundamental matrix for \(\mathbf{x}^{\prime}=A \mathbf{x}\) and hence, find \(e^{A t}\). $$A=\left[\begin{array}{ll}2 & 1 \\\0 & 2\end{array}\right]$$.

Use the variation-of-parameters technique to find a particular solution \(\mathbf{x}_{p}\) to \(\mathbf{x}^{\prime}=A \mathbf{x}+\mathbf{b},\) for the given \(A\) and \(\mathbf{b} .\) Also obtain the general solution to the system of differential equations. $$A=\left[\begin{array}{rr} 2 & -1 \\ -1 & 2 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{c} 0 \\ 4 e^{t} \end{array}\right]$$

Show that the given vector functions are linearly independent on \((-\infty, \infty)\). $$\mathbf{x}_{1}(t)=\left[\begin{array}{l} t \\ t \end{array}\right], \quad \mathbf{x}_{2}(t)=\left[\begin{array}{l} t \\ t^{2} \end{array}\right]$$

Solve the given system of differential equations. $$x_{1}^{\prime}=2 x_{1}-3 x_{2}, \quad x_{2}^{\prime}=x_{1}-2 x_{2}$$

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). $$\left[\begin{array}{rr} -6 & 1 \\ 6 & -5 \end{array}\right]$$

Show that the given functions are solutions of the system \(\mathbf{x}^{\prime}(t)=A(x) \mathbf{x}(t)\) for the given matrix \(A,\) and hence, find the general solution to the system (remember to check linear independence). If auxiliary conditions are given, find the particular solution that satisfies these conditions. $$\mathbf{x}_{1}(t)=\left[\begin{array}{l} e^{-t} \cos 2 t \\ e^{-t} \sin 2 t \end{array}\right], \quad \mathbf{x}_{2}(t)=\left[\begin{array}{c} -e^{-t} \sin 2 t \\ e^{-t} \cos 2 t \end{array}\right]$$, $$A=\left[\begin{array}{ll} 1 & -2 \\ 2 & 1 \end{array}\right], \quad \mathbf{x}(0)=\left[\begin{array}{l} 1 \\ 3 \end{array}\right]$$.

Show that the given vector functions are linearly independent on \((-\infty, \infty)\). $$\begin{array}{l} \mathbf{x}_{1}(t)=\left[\begin{array}{c} t+1 \\ t-1 \\ 2 t \end{array}\right], \quad \mathbf{x}_{2}(t)=\left[\begin{array}{c} e^{t} \\ e^{2 t} \\ e^{3 t} \end{array}\right] \\ \mathbf{x}_{3}(t)=\left[\begin{array}{c} 1 \\ \sin t \\ \cos t \end{array}\right] \end{array}$$

Determine the general solution to the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\) $$\left[\begin{array}{rr} -3 & -2 \\ 2 & 1 \end{array}\right]$$

Determine all equilibrium points of the given system and, if possible, characterize them as centers, spirals, saddles, or nodes. $$x^{\prime}=x-y^{2}, \quad y^{\prime}=y(9 x-4)$$

Use the techniques from Section 9.4 and Section 9.5 to determine a fundamental matrix for \(\mathbf{x}^{\prime}=A \mathbf{x}\) and hence, find \(e^{A t}\). $$A=\left[\begin{array}{rr}1 & 2 \\\0 & -1\end{array}\right]$$.

Use the variation-of-parameters technique to find a particular solution \(\mathbf{x}_{p}\) to \(\mathbf{x}^{\prime}=A \mathbf{x}+\mathbf{b},\) for the given \(A\) and \(\mathbf{b} .\) Also obtain the general solution to the system of differential equations. $$A=\left[\begin{array}{ll} 3 & 1 \\ 0 & 3 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l} t e^{3 t} \\ e^{3 t} \end{array}\right]$$

Solve the given system of differential equations. $$x_{1}^{\prime}=4 x_{1}+2 x_{2}, \quad x_{2}^{\prime}=-x_{1}+x_{2}$$

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). $$\left[\begin{array}{ll} 9 & -2 \\ 5 & -2 \end{array}\right]$$

Determine the general solution to the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\) $$\left[\begin{array}{rrr} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & -1 \end{array}\right]$$

Determine all equilibrium points of the given system and, if possible, characterize them as centers, spirals, saddles, or nodes. $$x^{\prime}=x+3 y^{2}, \quad y^{\prime}=y(x-2)$$

Show that the given functions are solutions of the system \(\mathbf{x}^{\prime}(t)=A(x) \mathbf{x}(t)\) for the given matrix \(A,\) and hence, find the general solution to the system (remember to check linear independence). If auxiliary conditions are given, find the particular solution that satisfies these conditions. $$\mathbf{x}_{1}(t)=\left[\begin{array}{c} e^{2 t} \\ -e^{2 t} \end{array}\right], \quad \mathbf{x}_{2}(t)=\left[\begin{array}{c} e^{2 t}(1+t) \\ -t e^{2 t} \end{array}\right]$$. $$A=\left[\begin{array}{rr} 3 & 1 \\\ -1 & 1 \end{array}\right]$$.

Use the techniques from Section 9.4 and Section 9.5 to determine a fundamental matrix for \(\mathbf{x}^{\prime}=A \mathbf{x}\) and hence, find \(e^{A t}\). $$A=\left[\begin{array}{rrr}3 & 0 & 0 \\\0 & 3 & -1 \\\0 & 1 & 1\end{array}\right]$$.

Use the variation-of-parameters technique to find a particular solution \(\mathbf{x}_{p}\) to \(\mathbf{x}^{\prime}=A \mathbf{x}+\mathbf{b},\) for the given \(A\) and \(\mathbf{b} .\) Also obtain the general solution to the system of differential equations. $$A=\left[\begin{array}{rr} -1 & 1 \\ 3 & 1 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{c} 20 e^{3 t} \\ 12 e^{t} \end{array}\right]$$

Show that the given vector functions are linearly independent on \((-\infty, \infty)\). $$\mathbf{x}_{1}(t)=\left[\begin{array}{l} t \\ t \end{array}\right], \quad \mathbf{x}_{2}(t)=\left[\begin{array}{l} |t| \\ t \end{array}\right]$$

Solve the given system of differential equations. $$x_{1}^{\prime}=2 x_{1}+4 x_{2}, \quad x_{2}^{\prime}=-4 x_{1}-6 x_{2}$$

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). $$\left[\begin{array}{rr} 10 & -4 \\ 4 & 2 \end{array}\right]$$

Determine the general solution to the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\) $$\left[\begin{array}{lll} 2 & 2 & -1 \\ 2 & 1 & -1 \\ 2 & 3 & -1 \end{array}\right]$$

Determine all equilibrium points of the given system and, if possible, characterize them as centers, spirals, saddles, or nodes. $$x^{\prime}=2 x+5 y^{2}, \quad y^{\prime}=y(3-4 x)$$

Characterize the equilibrium point for the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) and sketch the phase portrait. $$A=\left[\begin{array}{rr} 1 & 3 \\ 1 & -1 \end{array}\right]$$

Find \(n\) linearly independent solutions to \(\mathbf{x}^{\prime}=A \mathbf{x}\) of the form \(e^{A t} \mathbf{v},\) and hence find \(e^{A t}\). $$A=\left[\begin{array}{rr}-3 & -2 \\\2 & 1\end{array}\right]$$.

Use the variation-of-parameters technique to find a particular solution \(\mathbf{x}_{p}\) to \(\mathbf{x}^{\prime}=A \mathbf{x}+\mathbf{b},\) for the given \(A\) and \(\mathbf{b} .\) Also obtain the general solution to the system of differential equations. $$A=\left[\begin{array}{ll} -1 & 2 \\ -2 & 4 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{c} 54 t e^{3 t} \\ 9 e^{3 t} \end{array}\right]$$

Is there an interval on which \(\mathbf{x}_{1}(t)\) and \(\mathbf{x}_{2}(t)\) in this exercise are not linearly independent? $$\begin{aligned} &\mathbf{x}_{1}(t)=\left[\begin{array}{c} \sin t \\ \cos t \\ 1 \end{array}\right], \quad \mathbf{x}_{2}(t)=\left[\begin{array}{c} t \\ 1-t \\ 1 \end{array}\right]\\\ &\mathbf{x}_{3}(t)=\left[\begin{array}{c} \sinh t \\ \cosh t \\ 1 \end{array}\right] \end{aligned}$$

Solve the given system of differential equations. $$x_{1}^{\prime}=2 x_{2}, \quad x_{2}^{\prime}=-2 x_{1}$$

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). $$\left[\begin{array}{cc} -8 & 5 \\ -5 & 2 \end{array}\right]$$

Determine the general solution to the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\) $$\left[\begin{array}{rrr} -2 & 0 & 0 \\ 1 & -3 & -1 \\ -1 & 1 & -1 \end{array}\right]$$

Characterize the equilibrium point for the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) and sketch the phase portrait. $$A=\left[\begin{array}{rr} 0 & 2 \\ -2 & 0 \end{array}\right]$$

Find \(n\) linearly independent solutions to \(\mathbf{x}^{\prime}=A \mathbf{x}\) of the form \(e^{A t} \mathbf{v},\) and hence find \(e^{A t}\). $$A=\left[\begin{array}{ll}3 & -1 \\\4 & -1\end{array}\right]$$.

Use the variation-of-parameters technique to find a particular solution \(\mathbf{x}_{p}\) to \(\mathbf{x}^{\prime}=A \mathbf{x}+\mathbf{b},\) for the given \(A\) and \(\mathbf{b} .\) Also obtain the general solution to the system of differential equations. $$A=\left[\begin{array}{rr} 2 & 4 \\ -2 & -2 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l} 8 \sin 2 t \\ 8 \cos 2 t \end{array}\right]$$

Show that the given vector functions are linearly dependent on \((-\infty, \infty)\). $$\mathbf{x}_{1}(t)=\left[\begin{array}{c} e^{t} \\ 2 e^{2 t} \end{array}\right], \quad \mathbf{x}_{2}(t)=\left[\begin{array}{c} 4 e^{t} \\ 8 e^{2 t} \end{array}\right]$$

Solve the given system of differential equations. $$x_{1}^{\prime}=x_{1}-3 x_{2}, \quad x_{2}^{\prime}=3 x_{1}+x_{2}$$

Show that the given functions are solutions of the system \(\mathbf{x}^{\prime}(t)=A(x) \mathbf{x}(t)\) for the given matrix \(A,\) and hence, find the general solution to the system (remember to check linear independence). If auxiliary conditions are given, find the particular solution that satisfies these conditions. $$\mathbf{x}_{1}(t)=\left[\begin{array}{c} t \sin t \\ \cos t \end{array}\right], \quad \mathbf{x}_{2}(t)=\left[\begin{array}{c} -t \cos t \\\ \sin t \end{array}\right]$$, $$A=\left[\begin{array}{rr} 1 / t & t \\ -1 / t & 0 \end{array}\right]$$.

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). $$\left[\begin{array}{rrr} 3 & 0 & 4 \\ 0 & 2 & 0 \\ -4 & 0 & -5 \end{array}\right]$$

Determine the general solution to the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\) $$\left[\begin{array}{rrr} 15 & -32 & 12 \\ 8 & -17 & 6 \\ 0 & 0 & -1 \end{array}\right]$$