Chapter 9

Characterize the equilibrium point for the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) and sketch the phase portrait. $$A=\left[\begin{array}{ll} 1 & 0 \\ 3 & 1 \end{array}\right]$$

Find \(n\) linearly independent solutions to \(\mathbf{x}^{\prime}=A \mathbf{x}\) of the form \(e^{A t} \mathbf{v},\) and hence find \(e^{A t}\). $$A=\left[\begin{array}{rrr}2 & 0 & 0 \\\0 & 1 & -8 \\\0 & 2 & -7\end{array}\right]$$.

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. $$A=\left[\begin{array}{rr} 3 & 2 \\ -2 & -1 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{c} -3 e^{t} \\ 6 t e^{t} \end{array}\right]$$

Show that the given vector functions are linearly dependent on \((-\infty, \infty)\). $$\mathbf{x}_{1}(t)=\left[\begin{array}{c} t^{2} \\ 6-t+t^{3} \end{array}\right], \quad \mathbf{x}_{2}(t)=\left[\begin{array}{c} -3 t^{2} \\ -18 t+3 t^{2}-3 t^{3} \end{array}\right]$$

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

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. $$A=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 2 & -3 & 2 \\ 1 & -2 & 2 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{c} -e^{t} \\ 6 e^{-t} \\ e^{t} \end{array}\right]$$

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). $$\left[\begin{array}{rrr} -3 & -1 & 0 \\ 4 & -7 & 0 \\ 6 & 6 & 4 \end{array}\right]$$

Determine the general solution to the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\) $$\left[\begin{array}{lll} 4 & 0 & 0 \\ 1 & 4 & 0 \\ 0 & 1 & 4 \end{array}\right]$$

Determine all equilibrium points of the given system and, if possible, characterize them as centers, spirals, saddles, or nodes. $$x^{\prime}=x(1-y), \quad y^{\prime}=y(x+1)$$

If \(\mathbf{x}_{1}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{n}\) are solutions to \(\mathbf{x}^{\prime}=A(t) \mathbf{x}\) and \(X=\left[\mathbf{x}_{1}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{n}\right],\) prove that $$ X^{\prime}=A(t) X $$.

Characterize the equilibrium point for the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) and sketch the phase portrait. $$A=\left[\begin{array}{rr} 2 & 3 \\ -1 & -2 \end{array}\right]$$

Solve \(\mathbf{x}^{\prime}=A \mathbf{x}\) by determining \(n\) linearly independent solutions of the form \(\mathbf{x}(t)=e^{A t} \mathbf{v}\). \(A=\left[\begin{array}{rrr}0 & 1 & 3 \\ 2 & 3 & -2 \\ 1 & 1 & 2\end{array}\right] .\) You may assume that \(p(\lambda)=\) \(-(\lambda+1)(\lambda-3)^{2}\).

Show that the given vector functions are linearly dependent on \((-\infty, \infty)\). $$\begin{aligned} &\mathbf{x}_{1}(t)=\left[\begin{array}{c} t \\ t^{2} \\ -t^{3} \end{array}\right], \quad \mathbf{x}_{2}(t)=\left[\begin{array}{c} 2 t \\ 3 t^{2} \\ 0 \end{array}\right]\\\ &\mathbf{x}_{3}(t)=\left[\begin{array}{c} -t \\ 0 \\ 3 t^{3} \end{array}\right] \end{aligned}$$

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

Show that the given vector functions are linearly dependent on \((-\infty, \infty)\). $$\begin{aligned} &\mathbf{x}_{1}(t)=\left[\begin{array}{c} \sin ^{2} t \\ \cos ^{2} t \\ 2 \end{array}\right], \quad \mathbf{x}_{2}(t)=\left[\begin{array}{c} 2 \cos ^{2} t \\ 2 \sin ^{2} t \\ 1 \end{array}\right]\\\ &\mathbf{x}_{3}(t)=\left[\begin{array}{l} 2 \\ 2 \\ 5 \end{array}\right] \end{aligned}$$

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). $$\left[\begin{array}{rr} 3 & 13 \\ -1 & -3 \end{array}\right]$$

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

Let \(X(t)\) be a fundamental matrix for \(\mathbf{x}^{\prime}=A(t) \mathbf{x}\) on the interval \(I.\) (a) Show that the general solution to the linear system can be written as $$ \mathbf{x}=X(t) \mathbf{c}, $$ where \(c\) is a vector of constants. (b) If \(t_{0} \in I,\) show that the solution to the initial-value problem $$ \mathbf{x}^{\prime}=A \mathbf{x}, \quad \mathbf{x}\left(t_{0}\right)=\mathbf{x}_{0} $$ can be written as $$ \mathbf{x}=X(t) X^{-1}\left(t_{0}\right) \mathbf{x}_{0} $$.

Characterize the equilibrium point for the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) and sketch the phase portrait. $$A=\left[\begin{array}{rr} -2 & 3 \\ -3 & -2 \end{array}\right]$$

Solve \(\mathbf{x}^{\prime}=A \mathbf{x}\) by determining \(n\) linearly independent solutions of the form \(\mathbf{x}(t)=e^{A t} \mathbf{v}\). \(A=\left[\begin{array}{rrr}-8 & 6 & -3 \\ -12 & 10 & -3 \\ -12 & 12 & -2\end{array}\right] .\) You may assume that \(p(\lambda)=\)\ \(-(\lambda+2)^{2}(\lambda-4)\).

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. $$A=\left[\begin{array}{rrr} -1 & -2 & 2 \\ 2 & 4 & -1 \\ 0 & 0 & 3 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{l} -e^{3 t} \\ 4 e^{3 t} \\ 3 e^{3 t} \end{array}\right]$$

Solve the given initial-value problem. $$x_{1}^{\prime}=2 x_{2}, \quad x_{2}^{\prime}=x_{1}+x_{2}, \quad x_{1}(0)=3, \quad x_{2}(0)=0$$

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). $$\left[\begin{array}{rr} -3 & -10 \\ 5 & 11 \end{array}\right]$$

Solve \(\mathbf{x}^{\prime}=A \mathbf{x}\) by determining \(n\) linearly independent solutions of the form \(\mathbf{x}(t)=e^{A t} \mathbf{v}\). \(A=\left[\begin{array}{rrrr}1 & 0 & 0 & 0 \\ 0 & 6 & -7 & 3 \\ 0 & 0 & 3 & -1 \\ 0 & -4 & 9 & -3\end{array}\right] .\) You may assume that \(p(\lambda)=(\lambda-1)(\lambda-2)^{3}.\)

Determine the general solution to the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\) $$\left[\begin{array}{rll} 3 & 1 & 0 \\ -1 & 5 & 0 \\ 0 & 0 & 4 \end{array}\right]$$

Characterize the equilibrium point for the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) and sketch the phase portrait. $$A=\left[\begin{array}{rr} -2 & 1 \\ 1 & -2 \end{array}\right]$$

Let \(X(t)\) be a fundamental matrix for the system \(\mathbf{x}^{\prime}=A(t) \mathbf{x}(t),\) where \(A(t)\) is an \(n \times n\) matrix function. Show that the solution to the initial-value problem $$\mathbf{x}^{\prime}(t)=A(t) \mathbf{x}(t)+\mathbf{b}(t), \quad \mathbf{x}\left(t_{0}\right)=\mathbf{x}_{0}$$ can be written as \(\mathbf{x}(t)=X(t) X^{-1}\left(t_{0}\right) \mathbf{x}_{0}+X(t) \int_{t_{0}}^{t} X^{-1}(s) \mathbf{b}(s) d s\)

Solve the given initial-value problem. $$\begin{aligned} &x_{1}^{\prime}=2 x_{1}+5 x_{2}, \quad x_{2}^{\prime}=-x_{1}-2 x_{2}\\\ &x_{1}(0)=0, x_{2}(0)=1 \end{aligned}$$

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). $$\left[\begin{array}{rrr} -1 & -5 & 1 \\ 4 & -9 & -1 \\ 0 & 0 & 3 \end{array}\right]$$

Consider the nonhomogeneous system $$\begin{array}{l} x_{1}^{\prime}=2 x_{1}-3 x_{2}+34 \sin t \\ x_{2}^{\prime}=-4 x_{1}-2 x_{2}+17 \cos t \end{array}$$ Find the general solution to this system by first solving the associated homogeneous system, and then using the method of undetermined coefficients to obtain a particular solution. [Hint: The form of the nonhomogeneous term suggests a trial solution of the form $$\mathbf{x}_{p}(t)=\left[\begin{array}{l} A_{1} \cos t+B_{1} \sin t \\ A_{2} \cos t+B_{2} \sin t \end{array}\right]$$ where the constants \(A_{1}, A_{2}, B_{1},\) and \(B_{2}\) can be determined by substituting into the given system.]

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

The matrix \(A=\left[\begin{array}{rrrr}0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & 1 & 0\end{array}\right]\) has characteristic polynomial \(p(\lambda)=\left(\lambda^{2}+1\right)^{2} .\) Determine two complex-valued solutions to \(\mathbf{x}^{\prime}=A \mathbf{x}\) of the form \(\mathbf{x}=e^{A t} \mathbf{v},\) and hence, find four linearly independent real-valued solutions to the differential system.

Characterize the equilibrium point for the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) and sketch the phase portrait. $$A=\left[\begin{array}{ll} 5 & 4 \\ 4 & 5 \end{array}\right]$$

Let \(A(t)\) be an \(n \times n\) matrix function. Prove that the set of all solutions \(\mathbf{x}\) to the system \(\mathbf{x}^{\prime}(t)=A(t) \mathbf{x}(\mathrm{t})\) is a subspace of \(V_{n}(I)\)

Solve the given initial-value problem. $$\begin{aligned} &x_{1}^{\prime}=2 x_{1}+x_{2}, \quad x_{2}^{\prime}=-x_{1}+4 x_{2}\\\ &x_{1}(0)=1, x_{2}(0)=3 \end{aligned}$$

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). $$\left[\begin{array}{rrr} -4 & 0 & 0 \\ 2 & 5 & -9 \\ 0 & 5 & -1 \end{array}\right]$$

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

Characterize the equilibrium point for the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) and sketch the phase portrait. $$A=\left[\begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array}\right]$$

If \(A=\left[\begin{array}{ll}2 & -4 \\ 1 & -3\end{array}\right],\) determine two linearly independent solutions to \(\mathbf{x}^{\prime}=A \mathbf{x}\) on \((-\infty, \infty)\)

Solve the given non homogeneous system. $$x_{1}^{\prime}=x_{1}+2 x_{2}+5 e^{4 t}, \quad x_{2}^{\prime}=2 x_{1}+x_{2}$$

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). $$\left[\begin{array}{rrr} 2 & -2 & 1 \\ 1 & -4 & 1 \\ 2 & 2 & -3 \end{array}\right]$$ [Hint: The eigenvalues of \(A \text { are } \lambda=2,-2,-5 .]\)

Characterize the equilibrium point for the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) and sketch the phase portrait. $$A=\left[\begin{array}{ll} 3 & -2 \\ 2 & -1 \end{array}\right]$$

Problem 13 investigates the relationship between the Wronskian defined in this section for column vector functions in \(V_{n}(I)\) and the Wronskian defined previously for functions in \(C^{n}(I)\) Consider the differential equation $$ \frac{d^{2} y}{d t^{2}}+a \frac{d y}{d t}+b y=0 $$ where \(a\) and \(b\) are arbitrary functions of \(t\) (a) Show that Equation \((9.2 .4)\) can be replaced by the equivalent linear system $$ \mathbf{x}^{\prime}=A \mathbf{x} $$ where $$ A=\left[\begin{array}{rr} 0 & 1 \\ -b & -a \end{array}\right] \quad \text { and } \quad x_{1}=y, x_{2}=y^{\prime} $$ (b) If \(y_{1}=f_{1}(t)\) and \(y_{2}=f_{2}(t)\) are solutions to Equation (9.2.4) on an interval \(I,\) show that the corresponding solutions to the system (9.2.5) are $$ \mathbf{x}_{1}(t)=\left[\begin{array}{l} f_{1}(t) \\ f_{1}^{\prime}(t) \end{array}\right], \quad \mathbf{x}_{2}(t)=\left[\begin{array}{l} f_{2}(t) \\ f_{2}^{\prime}(t) \end{array}\right] $$ c) Show that $$ W\left[\mathbf{x}_{1}, \mathbf{x}_{2}\right](t)=W\left[y_{1}, y_{2}\right](t) $$

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

Solve the given non homogeneous system. $$x_{1}^{\prime}=-2 x_{1}+x_{2}+t, \quad x_{2}^{\prime}=-2 x_{1}+x_{2}+1$$

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). \(\left[\begin{array}{rrr}2 & -4 & 3 \\ -9 & -3 & -9 \\ 4 & 4 & 3\end{array}\right]\) [Hint: The eigenvalues of \(A \text { are } \lambda=6,-3,-1 .]\)

Characterize the equilibrium point for the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) and sketch the phase portrait. $$A=\left[\begin{array}{rr} 2 & -1 \\ 1 & 2 \end{array}\right]$$

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

Solve the given non homogeneous system. $$x_{1}^{\prime}=x_{1}+x_{2}+e^{2 t}, \quad x_{2}^{\prime}=3 x_{1}-x_{2}+5 e^{2 t}$$

Determine the general solution to the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). $$\left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \\ 4 & 5 & 6 & 7 \\ 7 & 6 & 5 & 4 \end{array}\right]$$