Chapter 9

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). \(\left[\begin{array}{rrr}-17 & 0 & -42 \\ -7 & 4 & -14 \\ 7 & 0 & 18\end{array}\right]\) [Hint: The eigenvalues of \(A \text { are } \lambda=4,-3 .]\)

Characterize the equilibrium point for the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) and sketch the phase portrait. $$A=\left[\begin{array}{ll} 2 & -5 \\ 4 & -7 \end{array}\right]$$

Solve the initial-value problem. $$ \begin{aligned} \mathbf{x}^{\prime}=A \mathbf{x}, \mathbf{x}(0) &=\mathbf{x}_{0}, \text { where } \\ A &=\left[\begin{array}{rr} -2 & -1 \\ 1 & -4 \end{array}\right], \quad \mathbf{x}_{0}=\left[\begin{array}{r} 0 \\ -1 \end{array}\right] \end{aligned} $$

Convert the given system of differential equations to a first-order linear system. $$\frac{d x}{d t}-t y=\cos t, \quad \frac{d^{2} y}{d t^{2}}-\frac{d x}{d t}+x=e^{t}$$

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). \(\left[\begin{array}{rrr}-16 & 30 & -18 \\ -8 & 8 & 16 \\ 8 & -15 & 9\end{array}\right]\) [Hint: The eigenvalues of \(A \text { are } \lambda=8,-7,0 .]\)

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

Characterize the equilibrium point for the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) and sketch the phase portrait. $$A=\left[\begin{array}{ll} 2 & 1 \\ 3 & 4 \end{array}\right]$$

Solve the initial-value problem. \(\mathbf{x}^{\prime}=A \mathbf{x}, \mathbf{x}(0)=\mathbf{x}_{0},\) where $$ A=\left[\begin{array}{rrr} -2 & -1 & 4 \\ 0 & -1 & 0 \\ -1 & -3 & 2 \end{array}\right], \quad \mathbf{x}_{0}=\left[\begin{array}{r} -2 \\ 1 \\ 1 \end{array}\right] $$

Convert the given system of differential equations to a first-order linear system. $$\frac{d^{2} x}{d t^{2}}-3 \frac{d y}{d t}+x=\sin t, \quad \frac{d^{2} y}{d t^{2}}-t \frac{d x}{d t}-e^{t} y=t^{2}$$

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). \(\left[\begin{array}{rrr}-7 & -6 & -7 \\ -3 & -3 & -3 \\ 7 & 6 & 7\end{array}\right]\) [Hint: The eigenvalues of \(A \text { are } \lambda=0,-3 .]\)

Solve the initial-value problem \(\mathbf{x}^{\prime}=\) \(A \mathbf{x}, \mathbf{x}(0)=\mathbf{x}_{0}\). $$A=\left[\begin{array}{rr} -1 & 4 \\ 2 & -3 \end{array}\right], \quad \mathbf{x}_{0}=\left[\begin{array}{l} 3 \\ 0 \end{array}\right]$$

Characterize the equilibrium point for the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) and sketch the phase portrait. $$A=\left[\begin{array}{rr} 3 & 4 \\ 4 & -3 \end{array}\right]$$

Show that if the vector differential equation \(\mathbf{x}^{\prime}=A \mathbf{x}\) has a solution of the form $$ \mathbf{x}(t)=e^{\lambda t}\left(\mathbf{v}_{2}+t \mathbf{v}_{1}+\frac{t^{2}}{2 !} \mathbf{v}_{2}\right) $$ then $$ \begin{array}{c} (A-\lambda I) \mathbf{v}_{0}=\mathbf{0}, \quad(A-\lambda I) \mathbf{v}_{1}=\mathbf{v}_{0}, \quad \text { and } \\ (A-\lambda I) \mathbf{v}_{2}=\mathbf{v}_{1} \end{array} $$

Convert the given linear differential equations to a first-order linear system. $$y^{\prime \prime}+2 t y^{\prime}+y=\cos t$$

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). \(\left[\begin{array}{rrr}3 & -1 & -2 \\ 1 & 6 & 1 \\ 1 & 0 & 6\end{array}\right]\) [Hint: The only eigenvalue of \(A \text { is } \lambda=5 .]\)

Solve the initial-value problem \(\mathbf{x}^{\prime}=\) \(A \mathbf{x}, \mathbf{x}(0)=\mathbf{x}_{0}\). $$A=\left[\begin{array}{rr} -1 & -6 \\ 3 & 5 \end{array}\right], \quad \mathbf{x}_{0}=\left[\begin{array}{l} 2 \\ 2 \end{array}\right]$$

Characterize the equilibrium point for the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) and sketch the phase portrait. $$A=\left[\begin{array}{rr} 1 & 1 \\ -9 & -5 \end{array}\right]$$

Let \(A\) be a \(2 \times 2\) real matrix. Prove that all solutions to \(\mathbf{x}^{\prime}=A \mathbf{x}\) satisfy $$ \lim _{t \rightarrow \infty} \mathbf{x}(t)=\mathbf{0} $$ if and only if all eigenvalues of \(A\) have negative real part.

Convert the given linear differential equations to a first-order linear system. $$y^{\prime \prime}+a y^{\prime}+b y=F(t), \quad a, b \text { constants. }$$

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). \(\left[\begin{array}{rrr}-1 & -4 & -2 \\ -4 & -5 & -6 \\ 4 & 8 & 7\end{array}\right]\) [Hint: The eigenvalues of \(A \text { are } \lambda=-1,1 \pm 2 i .]\)

Convert the given linear differential equations to a first-order linear system. $$y^{\prime \prime \prime}+t^{2} y^{\prime}-e^{t} y=t$$

Solve the initial-value problem \(\mathbf{x}^{\prime}=\) \(A \mathbf{x}, \mathbf{x}(0)=\mathbf{x}_{0}\). $$A=\left[\begin{array}{rrr} 2 & -1 & 3 \\ 3 & 1 & 0 \\ 2 & -1 & 3 \end{array}\right], \quad \mathbf{x}_{0}=\left[\begin{array}{r} -4 \\ 4 \\ 4 \end{array}\right]$$

Consider the predator-prey model $$\frac{d x}{d t}=x(2-y), \frac{d y}{d t}=y(x-2)$$ Sketch the phase plane for \(0 \leq x \leq 10,0 \leq y \leq 10\) Compare the behavior of the two specific cases corresponding to the initial conditions \(x(0)=1, y(0)=\) \(0.1,\) and \(x(0)=1, y(0)=1\)

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

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). \(\left[\begin{array}{rrr}7 & -2 & 2 \\ 0 & 4 & -1 \\ -1 & 1 & 4\end{array}\right]\) [Hint: The only eigenvalue of \(A \text { is } \lambda=5 .]\)

Consider the predator-prey model $$\frac{d x}{d t}=x(3-x-y), \frac{d y}{d t}=y(x-1)$$ Sketch the phase plane for \(0 \leq x \leq 4,0 \leq y \leq 4\) What happens to the populations of both species as \(t \rightarrow+\infty ?\)

Solve the initial-value problem $$ \mathbf{x}^{\prime}=A \mathbf{x}, \quad \mathbf{x}(0)=\left[\begin{array}{l} 1 \\ 1 \end{array}\right] $$ when \(A=\left[\begin{array}{rl}0 & 4 \\ -4 & 0\end{array}\right]\). Sketch the solution in the \(x_{1} x_{2}\) plane.

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

This problem outlines a proof of Theorem 9.5 .4 using results from the optional section on Jordan canonical forms, Section 7.6. (a) Conclude from the summary preceding Example 7.6 .11 that there are \(r\) cycles of generalized eigen vectors of \(A\) corresponding to \(\lambda\). Let the lengths of these cycles be \(l_{1}, l_{2}, \ldots, l_{r},\) respectively. (b) How are \(k_{i}\) and \(l_{i}\) related for each \(i ?\) Show that \(k_{i} \geq 0\) for each \(i\) and that \(k_{1}+k_{2}+\cdots+k_{r}=\) \(m-r\) (c) Conclude that for each \(i\) we have a cycle of generalized eigen vectors \(\left\\{\mathbf{v}_{0}^{(i)}, \mathbf{v}_{1}^{(i)}, \ldots, \mathbf{v}_{k_{i}}^{(i)}\right\\}\) satisfying (9.5.15)-(9.5.19). (d) By Theorem \(7.6 .10,\) the vectors in the cycle in part (c) are linearly independent. Conclude that the corresponding vector functions in (9.5.11)(9.5.14) are linearly independent. (e) Show that the functions defined in (9.5.11)(9.5.14) whose terms satisfy \((9.5 .15)-(9.5 .19)\) are solutions to the vector differential equation \(\mathbf{x}^{\prime}=A \mathbf{x} .\) This proves part (1) of Theorem 9.5 .4 (f) Deduce part ( 2 ) of Theorem 9.5 .4

The initial-value problem that governs the behavior of a coupled spring-mass system is (see the introduction to this chapter) $$ \begin{aligned} m_{1} \frac{d^{2} x}{d t^{2}} &=-k_{1} x+k_{2}(y-x) \\ m_{2} \frac{d^{2} y}{d t^{2}} &=-k_{2}(y-x) \\ x(0)=\alpha_{1}, & x^{\prime}(0)=\alpha_{2}, \quad y(0)=\alpha_{3}, \quad y^{\prime}(0)=\alpha_{4} \end{aligned} $$ where \(\alpha_{1}, \alpha_{2}, \alpha_{3},\) and \(\alpha_{4}\) are constants. Convert this problem into an initial-value problem for an equivalent first- order linear system. (You must give the appropriate initial conditions in the new variables.)

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

Consider the differential equation $$ \frac{d^{2} x}{d t^{2}}+b \frac{d x}{d t}+c x=0 $$ where \(b\) and \(c\) are constants. (a) Show that Equation (9.4.5) can be replaced by the equivalent first-order linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) where \(A=\left[\begin{array}{rr}0 & 1 \\ -c & -b\end{array}\right]\) (b) Show that the characteristic polynomial of \(A\) coincides with the auxiliary polynomial of Equation \((9.4 .5)\).

Consider the differential equation \(\frac{d^{2} y}{d t^{2}}+0.1(y-4)(y+1) \frac{d y}{d t}+y=0\) (a) Convert the differential equation to a first-order system using the substitution \(u=y, v=\frac{d y}{d t}\) and characterize the equilibrium point (0,0) (b) Sketch the phase plane for the system on the square \(-2 \leq u \leq 2,-2 \leq v \leq 2 .\) Based on the resulting sketch, do you think the differential equation has a limit cycle? (c) Repeat (b) using the square \(-8 \leq u \leq 8\) \(-8 \leq v \leq 8,\) and include the trajectories corresponding to the initial conditions \(u(0)=1\) \(v(0)=0,\) and \(u(0)=6, v(0)=0\)

Characterize the equilibrium point (0,0) for the sys\(\operatorname{tem} \mathbf{x}^{\prime}=A \mathbf{x}\) if \(A=\left[\begin{array}{rr}-1 & 2 \\ -2 & -1\end{array}\right] .\) Solve the system of differential equations, and show that the components of the solution vector satisfy $$ x^{2}+y^{2}=e^{-4 t}\left(c_{1}^{2}+c_{2}^{2}\right) $$ where \(c_{1}\) and \(c_{2}\) are constants. As \(t\) varies in Equation (9.9.9), describe the curve that is generated in the phase plane.

Solve the initial-value problem: $$ \begin{array}{c} x_{1}^{\prime}=-(\tan t) x_{1}+3 \cos ^{2} t \\ x_{2}^{\prime}=x_{1}+(\tan t) x_{2}+2 \sin t \\ x_{1}(0)=4, \quad x_{2}(0)=0 \end{array} $$

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

Let \(\lambda=a+i b, b \neq 0,\) be an eigenvalue of the \(n \times n\) (real) matrix \(A\) with corresponding eigenvector \(\mathbf{v}=\mathbf{r}+i \mathbf{s} .\) Then we have shown in the text that two real-valued solutions to \(\mathbf{x}^{\prime}=A \mathbf{x}\) are $$ \begin{array}{l} \mathbf{x}_{1}(t)=e^{a t}[\cos b t \mathbf{r}-\sin b t \mathbf{s}] \\ \mathbf{x}_{2}(t)=e^{a t}[\sin b t \mathbf{r}+\cos b t \mathbf{s}] \end{array} $$ Prove that \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\) are linearly independent on any interval. (You may assume that \(\mathbf{r}\) and \(\mathbf{s}\) are linearly independent in \(\mathbb{R}^{n} .\) ) The remaining problems in this section investigate general properties of solutions to \(\mathbf{x}^{\prime}=A \mathbf{x},\) where \(A\) is a nondefective matrix.

Convert the given differential equation to a first-order system using the substitution \(u=y, v=\frac{d y}{d t}\) and determine the phase portrait for the resulting system. $$\frac{d^{2} y}{d t^{2}}+6 \frac{d y}{d t}+9 y=0$$

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). $$\left[\begin{array}{rrrr} 2 & 13 & 0 & 0 \\ -1 & -2 & 0 & 0 \\ 0 & 0 & 2 & 4 \\ 0 & 0 & 0 & 2 \end{array}\right]$$

Let \(A\) be a \(2 \times 2\) nondefective matrix. If all eigenvalues of \(A\) have negative real part, prove that every solution to \(\mathbf{x}^{\prime}=A \mathbf{x}\) satisfies $$ \lim _{t \rightarrow \infty} \mathbf{x}(t)=\mathbf{0} $$

Convert the given differential equation to a first-order system using the substitution \(u=y, v=\frac{d y}{d t}\) and determine the phase portrait for the resulting system. $$\frac{d^{2} y}{d t^{2}}+16 y=0$$

Determine the general solution to the linear system \(\mathbf{x}^{\prime}=A \mathbf{x}\) for the given matrix \(A\). $$\left[\begin{array}{rrrr} 7 & 0 & 0 & -1 \\ 0 & 6 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 2 & 0 & 0 & 5 \end{array}\right]$$

Let \(A\) be a \(2 \times 2\) nondefective matrix. If every solution to \(\mathbf{x}^{\prime}=A \mathbf{x}\) satisfies \((9.4 .6),\) prove that all eigenvalues of \(A\) have negative real part.

Convert the given differential equation to a first-order system using the substitution \(u=y, v=\frac{d y}{d t}\) and determine the phase portrait for the resulting system. $$\frac{d^{2} y}{d t^{2}}+4 \frac{d y}{d t}+5 y=0$$

Use the variation-of-parameters method to determine a particular solution to the nonhomogeneous linear system \(\mathbf{x}^{\prime}=A \mathbf{x}+\mathbf{b} .\) Also find the general solution to the system. $$A=\left[\begin{array}{rr} -6 & 1 \\ 6 & -5 \end{array}\right], \mathbf{b}=\left[\begin{array}{c} 1 \\ e^{-t} \end{array}\right]$$

Determine the general solution to \(\mathbf{x}^{\prime}=A \mathbf{x}\) if \(A=\) \(\left[\begin{array}{rl}0 & b \\ -b & 0\end{array}\right],\) where \(b>0 .\) Describe the behavior of the solutions.

Convert the given differential equation to a first-order system using the substitution \(u=y, v=\frac{d y}{d t}\) and determine the phase portrait for the resulting system. $$\frac{d^{2} y}{d t^{2}}-25 y=0$$

Describe the behavior of the solutions to \(\mathbf{x}^{\prime}=A \mathbf{x},\) if \(A=\left[\begin{array}{rl}a & b \\ -b & a\end{array}\right],\) where \(a<0\) and \(b>0\).

Use the variation-of-parameters method to determine a particular solution to the nonhomogeneous linear system \(\mathbf{x}^{\prime}=A \mathbf{x}+\mathbf{b} .\) Also find the general solution to the system. $$A=\left[\begin{array}{ll} 9 & -2 \\ 5 & -2 \end{array}\right], \mathbf{b}=\left[\begin{array}{l} 9 t \\ 0 \end{array}\right]$$

Consider the differential equation $$ \frac{d^{2} y}{d t^{2}}+2 c \frac{d y}{d t}+k y=0 $$ where \(c\) and \(k\) are positive constants, that governs the behavior of a spring-mass system. Convert the differential equation to a first-order linear system and sketch the corresponding phase portraits. (You will need to distinguish the three cases \(c^{2}>k, c^{2}