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A Course in Mathematics for Students of Physics 1

Paul Bamberg, Shlomo Sternberg

Chapter 3

Linear differential equations in the plane - all with Video Answers

Educators

Chapter Questions

01:53

Problem 1

.(a) Write the power series expansions for $(1-X)^{-1}$ and for $(1-X)^{-2}$.
(b) Multiply these two series and compare the general term with the series for $(1-X)^{-3}$.

Adrian Co
Adrian Co
Numerade Educator

Problem 2

.(a) Let $F=\left(\begin{array}{ll}\frac{1}{4} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4}\end{array}\right)$. Prove that $F^2=\frac{1}{2} F$ and that $F^n=F / 2^{n-1}$. Using this result, evaluate the series expansion of $(I-F)^{-1}$. Compute the inverse directly, and compare.
(b) Try to evaluate $\left(\begin{array}{ll}\frac{3}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{3}{2}\end{array}\right)^{-1}$ by writing it as $(I+P)^{-1}$ where $P$ is the projection $\left(\begin{array}{cc}\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2}\end{array}\right)$ and using the series expansion of $(1+X)^{-1}$. Notice that although the inverse exists, the series fails to converge.

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Problem 3

(a) The matrix $N_{\pi / 4}=\left(\begin{array}{cc}-\frac{1}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{1}{2}\end{array}\right)$ has the property that $N_{\pi / 4}^2=0$. Taking advantage of this property, evaluate the matrix $F(t)=\exp \left(t N_{\pi / 4}\right)$ and check explicitly that $F^{\prime}(t)=N_{\pi / 4} F(t)$.
(b) The matrix $P_{\pi / 4}=\left(\begin{array}{cc}\frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2}\end{array}\right)$ has the property that $P_{\pi / 4}^2=P_{\pi / 4}$. Taking advantage of this property, evaluate $G(t)=\exp \left(t P_{\pi / 4}\right)$ and check that $G^{\prime}(t)=P_{\pi / 4} G(t)$.

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Problem 4

. Suppose that a matrix $P$ satisfies the equation $P^2=3 P$.
(a) What are the eigenvalues of $P$ ? Explain your reasoning.
(b) Using the power series for the exponential, show that $\exp (t P)$ can be expressed in the form

$$
\exp (t P)=I+g(t) P
$$

Find an expression for the function $g(t)$.

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Problem 5

Suppose that $B$ is a $2 \times 2$ matrix which has a repeated eigenvalue $\lambda$.
(a) Show that the matrix $N=B-\lambda I$ is nilpotent (i.e., $N^2=0$ ).
(b) By writing $B=N+\lambda I$ and using the series for the exponential function, show that

$$
\exp (t B)=(I+t N) \exp (t \lambda I)
$$

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Problem 6

Use exercise 3.5 to solve the system of equations

$$
\begin{aligned}
& \dot{x}(t)=x(t)-y(t) \\
& \dot{y}(t)=x(t)+3 y(t)
\end{aligned}
$$

for arbitrary initial conditions $\binom{x_0}{y_0}$.

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Problem 7

Calculate $\exp (t A)$ for the following matrices, and verify that $(\mathrm{d} / \mathrm{d} t)$ $\exp (t A)=A \exp (t A):$
(a) $A=\left(\begin{array}{ll}-4 & 5 \\ -2 & 3\end{array}\right)$.
(b) $A=\left(\begin{array}{ll}-1 & 9 \\ -1 & 5\end{array}\right)$. (Hint: $A=2 I+N$ where $N$ is nilpotent.)
(c) $A=\left(\begin{array}{ll}3 & -1 \\ 5 & -1\end{array}\right)$.

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14:21

Problem 8

Let $A$ be a $2 \times 2$ matrix which has two distinct real eigenvalues $\lambda_1$ and $\lambda_2$, with associated eigenvectors $v_1$ and $v_2$.
(a) Show that the matrix $P_1=\left(A-\lambda_2 I\right) /\left(\lambda_1-\lambda_2\right)$ is a projection onto the line determined by the eigenvector $\mathbf{v}_1: P_1^2=P_1$, the image of $P_1$ is the set of $\lambda \mathbf{v}_1$ and the kernel of $P_1$ is the set of $\lambda \mathbf{v}_2$.
(b) Similarly $P_2=\left(A-\lambda_1 I\right) /\left(\lambda_2-\lambda_1\right)$ is a projection onto the line determined by $\mathrm{v}_2$. Show that $P_1 P_2=P_2 P_1=0$, that $P_1+P_2=I$, and that $\lambda_1 P_1+\lambda_2 P_2=A$.
(c) By using the power series for the exponential, show that

$$
\exp \left(t \lambda_1 P_1+t \lambda_2 P_2\right)=\mathrm{e}^{\lambda_1 t} P_1+\mathrm{e}^{\lambda_2 t} P_2
$$

(d) Use this result to solve the equations

$$
\begin{aligned}
& \dot{x}(t)=-4 x(t)+5 y(t) \\
& \dot{y}(t)=-2 x(t)+3 y(t)
\end{aligned}
$$

for arbitrary initial conditions $\binom{x_0}{y_0}$.

Matthew Allcock
Matthew Allcock
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Problem 9

Let $A$ be a $2 \times 2$ matrix whose trace is 0 and whose determinant is 1 .
(a) Write down the characteristic equation of $A$, and state what this implies about $A^2$.
(b) Using the power series expansion of the exponential function, develop an expression for $\exp (t A)$ of the form

$$
\exp (t A)=F(t) I+G(t) A
$$

where $F(t)$ and $G(t)$ involve trigonometric functions of $t$.
(c) The solution curve for the equation $\dot{\mathbf{v}}=A \mathbf{v}$, with initial condition $\mathbf{v}=\mathbf{v}_0$, is an ellipse as shown in figure 3.19. Prove that all chords joining $\exp (t A) \mathbf{v}_0$ to $\exp (-t A) \mathbf{v}_0$ are parallel to $A \mathbf{v}_0$ and that the midpoint of each such chord lies on the diameter of the ellipse on which $\mathbf{v}_0$ lies.

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Problem 10

Suppose that $G$ is a matrix whose trace is zero and whose determinant is $-\beta^2$.
(a) According to the Cayley-Hamilton theorem, what does $G^2$ equal?
(b) Using the power series for the exponential function, show that $\exp G$ $+\exp (-G)$ is a multiple of the identity matrix. Find a function $f$ such that

$$
\exp (G)+\exp (-G)=f(\beta) I
$$

(c) By multiplying the above identity by $\exp G$ and applying the CayleyHamilton theorem, show that $\operatorname{Det}(\exp (G))=1$, and find an expression for the trace of $\exp G$.
(d) Let $F=\lambda I+G$. Using the above results, show that $\operatorname{Det}(\exp F)=\mathrm{e}^{(t+F)}$.

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02:02

Problem 11

For each of the following differential equations, determine which of the phase portraits given in cases 1 through 4 c best represents the nature of the general solution, then solve the equation completely for initial conditions $\binom{x_0}{y_{\mathrm{o}}}=\binom{3}{1}$ at $t=0$.
(a) $\dot{x}=-4 y$,

$$
\dot{y}=x-4 y
$$

(b)

$$
\begin{aligned}
& \dot{x}=x-2 y \\
& \dot{y}=-2 x+4 y
\end{aligned}
$$

(c)

$$
\begin{aligned}
& \dot{x}=4 x-5 y \\
& \dot{y}=4 x-4 y
\end{aligned}
$$

(d)

$$
\begin{aligned}
\dot{x} & =2 x+y \\
\dot{y} & =-x+4 y
\end{aligned}
$$

(e)

$$
\begin{aligned}
& \dot{x}=x-5 y \\
& \dot{y}=2 x-5 y \\
& \dot{x}=-2 x+4 y \\
& \dot{y}=-x+2 y
\end{aligned}
$$

(f)

Anand Jangid
Anand Jangid
Numerade Educator
02:02

Problem 12

For each of the following differential equations, determine which of the phase portraits given in cases 1 through 4 c best represents the general solution, then solve the equation completely for initial conditions $\binom{x_0}{y_0}=\binom{-2}{1}$ at $t=0$.
(a) $\dot{x}=3 y$,

$$
\dot{y}=x-2 y
$$

(b)

$$
\begin{aligned}
& \dot{x}=-x+y \\
& \dot{y}=-5 x+3 y
\end{aligned}
$$

(c)

$$
\begin{aligned}
\text { (c) } \dot{x} & =3 x+y, \\
\dot{y} & =-x+y . \\
\text { (d) } \dot{x} & =-5 x+4 y, \\
\dot{y} & =-8 x+7 y . \\
\text { (e) } \dot{x} & =-4 x-2 y, \\
\dot{y} & =5 x+2 y \\
\text { (f) } \dot{x} & =x+2 y, \\
\dot{y} & =2 x-4 y .
\end{aligned}
$$

(d)
(f)

Anand Jangid
Anand Jangid
Numerade Educator
04:19

Problem 13

By generalizing what you know about calculating and using the exponential of a $2 \times 2$ matrix to the $3 \times 3$ case, solve the differential equations

$$
\begin{aligned}
& \dot{x}=y \\
& \dot{y}=z \\
& \dot{z}=-6 x-11 y-6 z
\end{aligned}
$$

for initial conditions $\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\left(\begin{array}{r}1 \\ 2 \\ -1\end{array}\right)$ at $t=0$.
(Note: The one tricky new step is inverting a $3 \times 3$ matrix. If you regard this as the problem of solving three sets of simultaneous linear equations, you can do it by brute force.)

Tanishq Gupta
Tanishq Gupta
Numerade Educator
01:02

Problem 14

By generalizing the techniques which you already know. Solve the equations

$$
\dot{x}=x+y-z
$$
$$
\begin{aligned}
& \dot{y}=-x+5 y+z, \\
& \dot{z}=-2 x+2 y+4 z
\end{aligned}
$$

for initial conditions

$$
\left(\begin{array}{l}
x \\
y \\
z
\end{array}\right)=\left(\begin{array}{l}
1 \\
1 \\
1
\end{array}\right)
$$

Hast Aggarwal
Hast Aggarwal
Numerade Educator

Problem 15

By introducing the variable $v=\dot{x}$, convert the second-order differential equation

$$
\ddot{x}+4 \dot{x}+5 x=0
$$

to a pair of first-order equations, then solve these equations for arbitrary initial conditions $\binom{x_0}{v_0}$.

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Problem 16

(a) The differential equation for a critically damped harmonic oscillator, expressed in units chosen so that $\omega_0^2=1$, is

$$
\ddot{x}+2 \dot{x}+x=0 .
$$

Solve this equation by matrix methods, introducing $v=\dot{x}$ as a new variable. Write down the solution for initial conditions $\binom{x}{v}=\binom{x_0}{0}$ and for $\binom{x}{v}=\binom{0}{v_0}$, and sketch phase portraits of these and other solutions. Show that $x=0$ or $v=0$ can occur at most once.
(b) One way of solving the above equation without having to contend with a repeated eigenvalue is first to solve $\bar{x}+2 \dot{x}+\left(1-\varepsilon^2\right) x=0$, which leads to a matrix with distinct real eigenvalues, then let $\varepsilon \rightarrow 0$. (Physically, this corresponds to using a slightly weaker spring.) Carry through the procedure, first finding solutions for initial conditions $\binom{x_0}{0}$ and $\binom{0}{v_0}$, then letting $\varepsilon \rightarrow 0$. Show what happens to the phase portraits as $\varepsilon>0$.
(c) Another alternative is first to solve $\vec{x}+2 \dot{x}+\left(1+\varepsilon^2\right) x=0$, which leads to a matrix with complex eigenvalues, then let $\varepsilon \rightarrow 0$. Do this, again showing what happens to the solutions for initial conditions $\binom{x_0}{0}$ and $\binom{0}{v_0}$, and to the phase portraits, as $\varepsilon \rightarrow 0$.

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Problem 17

Consider the function $\cos t x$.
(a) Show, by use of formal power series, that

$$
\frac{\mathrm{d}^2}{\mathrm{~d} t^2}(\cos t x)=-x^2 \cos t x
$$

and that

$$
\frac{\mathrm{d}}{\mathrm{~d} t}(\cos t x)=0 \quad \text { for } \quad t=0
$$

(b) Suppose that $\binom{\ddot{x}}{\ddot{y}}=-B\binom{x}{y}$, where $B$ is a matrix which has a square root $A$. Show that $\cos t A\binom{x(0)}{y(0)}$ is a solution to the second-order system of equations

$$
\frac{\mathrm{d}^2}{\mathrm{~d} t^2} v(t)=-A^2 v(t)
$$

with initial conditions $v(0)=v_0$ and $\mathrm{d} v / \mathrm{d} t(0)=0$.
(c) Let $B=\left(\begin{array}{rr}\frac{5}{2} & -\frac{3}{2} \\ -\frac{3}{2} & \frac{5}{2}\end{array}\right)$. Find a matrix $A$, with positive eigenvalues, such that $A^2=B$. (Hint: diagonalize $B$.)
(d) For the matrix $A$ which you have just constructed, compute the matrix $\cos (t A)$. (Hint: You have already diagonalized $A$. Use procedure similar to that for computing $\exp (t A)$.)
(e) Use the above results to solve the equations

$$
\begin{aligned}
& \ddot{x}=-\frac{5}{2} x+\frac{3}{2} y \\
& \bar{y}=\frac{3}{2} x-\frac{5}{2} y
\end{aligned}
$$

for initial conditions $\dot{x}(0)=\dot{y}(0)=0, x(0)=x_0, y(0)=y_0$.

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02:04

Problem 18

Consider the system of differential equations

$$
\begin{aligned}
& \dot{x}=4 \beta x-y \\
& \dot{y}=9 x+\beta y
\end{aligned}
$$

where $\beta$ is a real-valued parameter.
(a) Solve the system for arbitrary initial conditions and $\beta=0$.
(b) Find two critical values of the parameter, $\beta_1<0$ and $\beta_2>0$, at which the nature of the solution changes. Discuss the solutions for $\beta=\beta_1$ and $\beta=\beta_2$.
(c) Draw phase portraits which describe qualitatively the nature of the solutions for $\beta<\beta_1, \beta_1<\beta<\beta_2$, and $\beta>\beta_2$.

Anand Jangid
Anand Jangid
Numerade Educator
01:35

Problem 19

Let $A=\left(\begin{array}{rr}-2 & 1 \\ 2 & -1\end{array}\right)$.
(a) Find matrices $D$ and $B$ so that $A=B D B^{-1}$.
(b) Construct the solution to the differential equation $\dot{\mathbf{v}}=A \mathbf{v}$ for arbitrary initial conditions $\mathbf{v}_0=\binom{x_0}{y_0}$ when $t=0$. Please remember that $\mathrm{e}^0=1$.
(c) Sketch a phase portrait for the equation $\dot{\mathbf{v}}=A \boldsymbol{v}$. Determine the image and kernel of the matrix

$$
F=\lim _{t \rightarrow \infty} \exp (A t)
$$

and explain their significance in relation to the phase portrait.
(d) By using the trial solution $\mathbf{v}=\exp (A t) \mathbf{w}$, construct a solution to the differential equation $\dot{\mathbf{v}}-A \mathbf{v}=\binom{1}{2}$.

A M
A M
Numerade Educator
02:18

Problem 20

.(a) By introducing $u=\dot{x}$ as a new variable, convert

$$
\ddot{x}+2 \dot{x}-3 x=3 \sin 2 t+2 \cos 2 t
$$

to an equation of the form

$$
\binom{\dot{x}}{\dot{u}}-A\binom{x}{u}=\mathbf{b}(t)
$$

(b) Solve this equation for initial conditions $x(0)=0, u(0)=0$ by using the results developed in section 3.4.

Victor Salazar
Victor Salazar
Numerade Educator
02:17

Problem 21

When an undamped oscillator is acted upon by a force at the natural frequency of the oscillator, conventional methods of solution fail because no steady state is ever achieved. The formula developed in the notes,

$$
\mathbf{v}(t)=\int_0^t \exp [(t-s) A] \mathbf{b}(s) \mathrm{d} s
$$

works fine, however. Use it to solve

$$
\binom{\dot{x}}{\dot{u}}-\left(\begin{array}{cc}
0 & 1 \\
-\omega^2 & 0
\end{array}\right)\binom{x}{u}=\binom{0}{\sin \omega t}
$$

for initial conditions $x(0)=0, u(0)=0$.

Surendra Kumar
Surendra Kumar
Numerade Educator