Content check — Differential Equations (Lebl), Chapter 4: Systems of ODEs

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 4: Systems of ODEs

37 problems · 1 figure

Exercise 3.1.101 §3.1 source ↗

Find the general solution to $y_{1}^{'} = 3 y_{1}$ ⁠, $y_{2}^{'} = y_{1} + y_{2}$ ⁠, $y_{3}^{'} = y_{1} + y_{3}$ ⁠.

Answer

$y_{1} = C_{1} e^{3 x}$ ⁠, $y_{2} = C_{2} e^{x} + \frac{C_{1}}{2} e^{3 x}$ ⁠, $y_{3} = C_{3} e^{x} + \frac{C_{1}}{2} e^{3 x}$

Exercise 3.1.102 §3.1 source ↗

Solve $y^{'} = 2 x$ ⁠, $x^{'} = x + y$ ⁠, $x (0) = 1$ ⁠, $y (0) = 3$ ⁠.

Answer

$x = \frac{5}{3} e^{2 t} - \frac{2}{3} e^{- t}$ ⁠, $y = \frac{5}{3} e^{2 t} + \frac{4}{3} e^{- t}$

Exercise 3.1.103 §3.1 source ↗

Write $x^{'''} = x + t$ as a first-order system.

Answer

$u_{1}^{'} = u_{2}$ ⁠, $u_{2}^{'} = u_{3}$ ⁠, $u_{3}^{'} = u_{1} + t$ ⁠, (here $u_{1} = x$ ⁠)

Exercise 3.1.104 §3.1 source ↗

Write $y_{1}^{''} + y_{1} + y_{2} = t$ ⁠, $y_{2}^{''} + y_{1} - y_{2} = t^{2}$ as a first-order system.

Answer

$y_{1}^{'} = y_{3}$ ⁠, $y_{2}^{'} = y_{4}$ ⁠, $y_{3}^{'} = - y_{1} - y_{2} + t$ ⁠, $y_{4}^{'} = - y_{1} + y_{2} + t^{2}$

Exercise 3.1.105 §3.1 source ↗

Suppose two masses on carts on frictionless surface are at displacements $x_{1}$ and $x_{2}$ as in Example 3.1.3. Suppose initial displacement is $x_{1} (0) = x_{2} (0) = 0$ ⁠, and initial velocity is $x_{1}^{'} (0) = x_{2}^{'} (0) = a$ for some number $a$ ⁠. Use your intuition to solve the system, explain your reasoning.

A diagram of two carts on wheels connected by a spring. The carts are labeled as m sub 1 and m sub 2 signifying the masses. The spring is labeled as k signifying the spring constant. There is a vertical tick and a rightward horizontal arrow below each cart labeled as x sub 1 and x sub 2 respectively, signifying that their state as shown is the rest position and x sub 1 and x sub 2 give the displacement of each cart from its rest position. — Example 3.1.3 in the book ↗

Answer

$x_{1} = x_{2} = a t$ ⁠. Explanation of the intuition is left to reader.

Exercise 3.1.106 §3.1 source ↗

Suppose the tanks are as in Example 3.1.2 except that clean water flows in at a rate of $s$ liters per second into tank 1, and brine flows out of tank 2 and into the sewer also at a rate of $s$ liters per second. The rate of flow from tank 1 into tank 2 is still $r$ ⁠, but the rate of flow from tank 2 back into tank 1 is $r - s$ (assume $r > s$ ⁠).

a) Draw the picture.

b) Set up the system of equations.

c) Intuitively, what happens as $t$ goes to infinity, explain.

A diagram of two tanks with liquid. The substance in the first tank is labeled as x sub 1 to signify the amount of salt and the substance in the second tank is labeled as x sub 2. Both tanks have a label that Vol. equals V to signify that the volume is the same and equal to V in both tanks. There are two pipes each carrying liquid from one tank to the other and they are both marked with r signifying the flow. — Example 3.1.2 in the book ↗

Answer

a) Left to reader. b) $x_{1}^{'} = \frac{r - s}{V} x_{2} - \frac{r}{V} x_{1}$ ⁠, $x_{2}^{'} = \frac{r}{V} (x_{1} - x_{2})$ ⁠. c) As $t$ goes to infinity, both $x_{1}$ and $x_{2}$ go to zero, explanation is left to reader.

Exercise 3.2.101 §3.2 source ↗

Compute determinant of $[\begin{matrix} 1 & 1 & 1 \\ 2 & 3 & - 5 \\ 1 & - 1 & 0 \end{matrix}]$

Answer

$- 15$

Exercise 3.2.102 §3.2 source ↗

Find $t$ such that $[\begin{matrix} 1 & t \\ - 1 & 2 \end{matrix}]$ is not invertible.

Answer

$- 2$

Exercise 3.2.103 §3.2 source ↗

Solve $[\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}] 𝐱 = [\begin{matrix} 10 \\ 20 \end{matrix}]$ ⁠.

Answer

$𝐱 = [\begin{matrix} 15 \\ - 5 \end{matrix}]$

Exercise 3.2.104 §3.2 source ↗

Suppose $a, b, c$ are nonzero numbers. Let $M = [\begin{matrix} a & 0 \\ 0 & b \end{matrix}]$ ⁠, $N = [\begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix}]$ ⁠.

a) Compute $M^{- 1}$ ⁠.

b) Compute $N^{- 1}$ ⁠.

Answer

a) $[\begin{matrix} ^{1} /_{a} & 0 \\ 0 & ^{1} /_{b} \end{matrix}]$ b) $[\begin{matrix} ^{1} /_{a} & 0 & 0 \\ 0 & ^{1} /_{b} & 0 \\ 0 & 0 & ^{1} /_{c} \end{matrix}]$

Exercise 3.3.101 §3.3 source ↗

Are $[\begin{matrix} e^{2 t} \\ e^{t} \end{matrix}]$ and $[\begin{matrix} e^{t} \\ e^{2 t} \end{matrix}]$ linearly independent? Justify.

Answer

Yes.

Exercise 3.3.102 §3.3 source ↗

Are $[\begin{matrix} \cosh (t) \\ 1 \end{matrix}]$ ⁠, $[\begin{matrix} e^{t} \\ 1 \end{matrix}]$ ⁠, and $[\begin{matrix} e^{- t} \\ 1 \end{matrix}]$ linearly independent? Justify.

Answer

No. $2 [\begin{matrix} \cosh (t) \\ 1 \end{matrix}] -$ $[\begin{matrix} e^{t} \\ 1 \end{matrix}] -$ $[\begin{matrix} e^{- t} \\ 1 \end{matrix}] =$ $𝟎$

Exercise 3.3.103 §3.3 source ↗

Write $x^{'} = 3 x - y + e^{t}$ ⁠, $y^{'} = t x$ in matrix notation.

Answer

${[\begin{matrix} x \\ y \end{matrix}]}^{'} = [\begin{matrix} 3 & - 1 \\ t & 0 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] + [\begin{matrix} e^{t} \\ 0 \end{matrix}]$

Exercise 3.3.104 §3.3 source ↗

a) Write $x_{1}^{'} = 2 t x_{2}$ ⁠, $x_{2}^{'} = 2 t x_{2}$ in matrix notation.

b) Solve and write the solution in matrix notation.

Answer

a) $𝐱^{'} = [\begin{matrix} 0 & 2 t \\ 0 & 2 t \end{matrix}] 𝐱$ b) $𝐱 = [\begin{matrix} C_{2} e^{t^{2}} + C_{1} \\ C_{2} e^{t^{2}} \end{matrix}]$

Exercise 3.4.101 §3.4 source ↗

a) Compute eigenvalues and eigenvectors of $A = [\begin{matrix} 1 & 0 & 3 \\ - 1 & 0 & 1 \\ 2 & 0 & 2 \end{matrix}]$ ⁠.

b) Solve the system $𝐱^{'} = A 𝐱$ ⁠.

Answer

a) Eigenvalues: $4,0, - 1$ Eigenvectors: $[\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}]$ ⁠, $[\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}]$ ⁠, $[\begin{matrix} 3 \\ 5 \\ - 2 \end{matrix}]$
b) $𝐱 =$ $C_{1} [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}] e^{4 t} +$ $C_{2} [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] +$ $C_{3} [\begin{matrix} 3 \\ 5 \\ - 2 \end{matrix}] e^{- t}$

Exercise 3.4.102 §3.4 source ↗

a) Compute eigenvalues and eigenvectors of $A = [\begin{matrix} 1 & 1 \\ - 1 & 0 \end{matrix}]$ ⁠.

b) Solve the system $𝐱^{'} = A 𝐱$ ⁠.

Answer

a) Eigenvalues: $\frac{1 + \sqrt{3} i}{2}, \frac{1 - \sqrt{3} i}{2}$ ⁠, Eigenvectors: $[\begin{matrix} - 2 \\ 1 - \sqrt{3} i \end{matrix}]$ ⁠, $[\begin{matrix} - 2 \\ 1 + \sqrt{3} i \end{matrix}]$
b) $𝐱 =$ $C_{1} e^{t / 2} [\begin{matrix} - 2 \cos (\frac{\sqrt{3} t}{2}) \\ \cos (\frac{\sqrt{3} t}{2}) + \sqrt{3} \sin (\frac{\sqrt{3} t}{2}) \end{matrix}] +$ $C_{2} e^{t / 2} [\begin{matrix} - 2 \sin (\frac{\sqrt{3} t}{2}) \\ \sin (\frac{\sqrt{3} t}{2}) - \sqrt{3} \cos (\frac{\sqrt{3} t}{2}) \end{matrix}]$

Exercise 3.4.103 §3.4 source ↗

Solve $x_{1}^{'} = x_{2}$ ⁠, $x_{2}^{'} = x_{1}$ using the eigenvalue method.

Answer

$𝐱 = C_{1} [\begin{matrix} 1 \\ 1 \end{matrix}] e^{t} + C_{2} [\begin{matrix} 1 \\ - 1 \end{matrix}] e^{- t}$

Exercise 3.4.104 §3.4 source ↗

Solve $x_{1}^{'} = x_{2}$ ⁠, $x_{2}^{'} = - x_{1}$ using the eigenvalue method.

Answer

$𝐱 =$ $C_{1} [\begin{matrix} \cos (t) \\ - \sin (t) \end{matrix}] +$ $C_{2} [\begin{matrix} \sin (t) \\ \cos (t) \end{matrix}]$

Exercise 3.5.101 §3.5 source ↗

Describe the behavior of the following systems without solving:

a) $x^{'} = x + y$ ⁠, $y^{'} = x - y$ ⁠.

b) $x_{1}^{'} = x_{1} + x_{2}$ ⁠, $x_{2}^{'} = 2 x_{2}$ ⁠.

c) $x_{1}^{'} = - 2 x_{2}$ ⁠, $x_{2}^{'} = 2 x_{1}$ ⁠.

d) $x^{'} = x + 3 y$ ⁠, $y^{'} = - 2 x - 4 y$ ⁠.

e) $x^{'} = x - 4 y$ ⁠, $y^{'} = - 4 x + y$ ⁠.

Answer

a) Two eigenvalues: $\pm \sqrt{2}$ so the behavior is a saddle. b) Two eigenvalues: $1$ and $2$ ⁠, so the behavior is a source. c) Two eigenvalues: $\pm 2 i$ ⁠, so the behavior is a center (ellipses). d) Two eigenvalues: $- 1$ and $- 2$ ⁠, so the behavior is a sink. e) Two eigenvalues: $5$ and $- 3$ ⁠, so the behavior is a saddle.

Exercise 3.5.102 §3.5 source ↗

Suppose that $𝐱^{'} = P 𝐱$ where $P$ is a 2 by 2 matrix with eigenvalues $2 \pm i$ ⁠. Describe the behavior.

Answer

Spiral source.

Exercise 3.5.103 §3.5 source ↗

Take ${[\begin{matrix} x \\ y \end{matrix}]}^{'} = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}]$ ⁠. Draw the vector field and describe the behavior. Is it one of the behaviors that we have seen before?

Answer (2 parts)

The solution does not move anywhere if $y = 0$ ⁠. When $y$ is positive, the solution moves (with constant speed) in the positive $x$ direction. When $y$ is negative, the solution moves (with constant speed) in the negative $x$ direction. It is not one of the behaviors we saw.
Note that the matrix has a double eigenvalue 0 and the general solution is $x = C_{1} t + C_{2}$ and $y = C_{1}$ ⁠, which agrees with the description above.

Exercise 3.6.101 §3.6 source ↗

Find the general solution to $[\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}] 𝐱^{''} =$ $[\begin{matrix} - 3 & 0 & 0 \\ 2 & - 4 & 0 \\ 0 & 6 & - 3 \end{matrix}] 𝐱 +$ $[\begin{matrix} \cos (2 t) \\ 0 \\ 0 \end{matrix}]$ ⁠.

Answer

$𝐱 =$ $[\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix}] (a_{1} \cos (\sqrt{3} t) +$ $b_{1} \sin (\sqrt{3} t)) +$ $[\begin{matrix} 0 \\ 1 \\ - 2 \end{matrix}] (a_{2} \cos (\sqrt{2} t) +$ $b_{2} \sin (\sqrt{2} t)) +$
$[\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] (a_{3} \cos (t) +$ $b_{3} \sin (t)) +$ $[\begin{matrix} - 1 \\ ^{1} /_{2} \\ ^{- 1} /_{3} \end{matrix}] \cos (2 t)$

Exercise 3.6.102 §3.6 source ↗

Suppose there are three carts of equal mass $m$ and connected by two springs of constant $k$ (and no connections to walls). Set up the system and find its general solution.

Answer

$[\begin{matrix} m & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & m \end{matrix}] 𝐱^{''} =$ $[\begin{matrix} - k & k & 0 \\ k & - 2 k & k \\ 0 & k & - k \end{matrix}] 𝐱$ ⁠. Solution: $𝐱 =$ $[\begin{matrix} 1 \\ - 2 \\ 1 \end{matrix}] (a_{1} \cos (\sqrt{^{3 k} /_{m}} t) +$ $b_{1} \sin (\sqrt{^{3 k} /_{m}} t)) +$ $[\begin{matrix} 1 \\ 0 \\ - 1 \end{matrix}] (a_{2} \cos (\sqrt{^{k} /_{m}} t) +$ $b_{2} \sin (\sqrt{^{k} /_{m}} t)) +$ $[\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] (a_{3} t + b_{3}) .$

Exercise 3.6.103 §3.6 source ↗

Suppose a cart of mass $2 kg$ is attached by a spring of constant $k = 1$ to a cart of mass $3 kg$ , which is attached to the wall by a spring also of constant $k = 1$ ⁠. Suppose that the initial position of the first cart is 1 meter in the positive direction from the rest position, and the second cart starts at the rest position. The carts are not moving and are let go. Find the position of the second cart as a function of time.

Answer

$x_{2} =$ $(^{2} /_{5}) \cos (\sqrt{^{1} /_{6}} t) -$ $(^{2} /_{5}) \cos (t)$

Exercise 3.7.101 §3.7 source ↗

Let $A = [\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}]$ ⁠.

a) What are the eigenvalues?

b) What is/are the defect(s) of the eigenvalue(s)?

c) Find the general solution of $𝐱^{'} = A 𝐱$ ⁠.

Answer

a) $3,0,0$ b) No defects. c) $𝐱 =$ $C_{1} [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] e^{3 t} +$ $C_{2} [\begin{matrix} 1 \\ 0 \\ - 1 \end{matrix}] +$ $C_{3} [\begin{matrix} 0 \\ 1 \\ - 1 \end{matrix}]$

Exercise 3.7.102 §3.7 source ↗

Let $A = [\begin{matrix} 1 & 3 & 3 \\ 1 & 1 & 0 \\ - 1 & 1 & 2 \end{matrix}]$ ⁠.

a) What are the eigenvalues?

b) What is/are the defect(s) of the eigenvalue(s)?

c) Find the general solution of $𝐱^{'} = A 𝐱$ ⁠.

Answer

a) $1,1,2$
b) Eigenvalue 1 has a defect of 1
c) $𝐱 =$ $C_{1} [\begin{matrix} 0 \\ 1 \\ - 1 \end{matrix}] e^{t} +$ $C_{2} ([\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] + t [\begin{matrix} 0 \\ 1 \\ - 1 \end{matrix}]) e^{t} +$ $C_{3} [\begin{matrix} 3 \\ 3 \\ - 2 \end{matrix}] e^{2 t}$

Exercise 3.7.103 §3.7 source ↗

Let $A = [\begin{matrix} 2 & 0 & 0 \\ - 1 & - 1 & 9 \\ 0 & - 1 & 5 \end{matrix}]$ ⁠.

a) What are the eigenvalues?

b) What is/are the defect(s) of the eigenvalue(s)?

c) Find the general solution of $𝐱^{'} = A 𝐱$ ⁠.

Answer

a) $2,2,2$
b) Eigenvalue 2 has a defect of 2
c) $𝐱 =$ $C_{1} [\begin{matrix} 0 \\ 3 \\ 1 \end{matrix}] e^{2 t} +$ $C_{2} ([\begin{matrix} 0 \\ - 1 \\ 0 \end{matrix}] + t [\begin{matrix} 0 \\ 3 \\ 1 \end{matrix}]) e^{2 t} +$ $C_{3} ([\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] +$ $t [\begin{matrix} 0 \\ - 1 \\ 0 \end{matrix}] +$ $\frac{t^{2}}{2} [\begin{matrix} 0 \\ 3 \\ 1 \end{matrix}]) e^{2 t}$

Exercise 3.7.104 §3.7 source ↗

Let $A = [\begin{matrix} a & a \\ b & c \end{matrix}]$ ⁠, where $a$ ⁠, $b$ ⁠, and $c$ are unknowns. Suppose that $5$ is a doubled eigenvalue of defect 1, and suppose that $[\begin{matrix} 1 \\ 0 \end{matrix}]$ is a corresponding eigenvector. Find $A$ and show that there is only one such matrix $A$ ⁠.

Answer

$A = [\begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}]$

Exercise 3.8.101 §3.8 source ↗

Compute $e^{t A}$ where $A = [\begin{matrix} 1 & - 2 \\ - 2 & 1 \end{matrix}]$ ⁠.

Answer

$e^{t A} =$ $[\begin{matrix} \frac{e^{3 t} + e^{- t}}{2} & \frac{e^{- t} - e^{3 t}}{2} \\ \frac{e^{- t} - e^{3 t}}{2} & \frac{e^{3 t} + e^{- t}}{2} \end{matrix}]$

Exercise 3.8.102 §3.8 source ↗

Compute $e^{t A}$ where $A = [\begin{matrix} 1 & - 3 & 2 \\ - 2 & 1 & 2 \\ - 1 & - 3 & 4 \end{matrix}]$ ⁠.

Answer

$e^{t A} =$ $[\begin{matrix} 2 e^{3 t} - 4 e^{2 t} + 3 e^{t} & \frac{3 e^{t}}{2} - \frac{3 e^{3 t}}{2} & - e^{3 t} + 4 e^{2 t} - 3 e^{t} \\ 2 e^{t} - 2 e^{2 t} & e^{t} & 2 e^{2 t} - 2 e^{t} \\ 2 e^{3 t} - 5 e^{2 t} + 3 e^{t} & \frac{3 e^{t}}{2} - \frac{3 e^{3 t}}{2} & - e^{3 t} + 5 e^{2 t} - 3 e^{t} \end{matrix}]$

Exercise 3.8.103 §3.8 source ↗

a) Compute $e^{t A}$ where $A = [\begin{matrix} 3 & - 1 \\ 1 & 1 \end{matrix}]$ ⁠.

b) Solve $𝐱^{'} = A 𝐱$ for $𝐱 (0) = [\begin{matrix} 1 \\ 2 \end{matrix}]$ ⁠.

Answer

a) $e^{t A} =$ $[\begin{matrix} (t + 1) e^{2 t} & - t e^{2 t} \\ t e^{2 t} & (1 - t) e^{2 t} \end{matrix}]$ b) $𝐱 = [\begin{matrix} (1 - t) e^{2 t} \\ (2 - t) e^{2 t} \end{matrix}]$

Exercise 3.8.104 §3.8 source ↗

Compute the first 3 terms (up to the second degree) of the Taylor expansion of $e^{t A}$ where $A = [\begin{matrix} 2 & 3 \\ 2 & 2 \end{matrix}]$ ⁠. Write it as a single matrix. Then use it to approximate $e^{0.1 A}$ ⁠.

Answer

$[\begin{matrix} 1 + 2 t + 5 t^{2} & 3 t + 6 t^{2} \\ 2 t + 4 t^{2} & 1 + 2 t + 5 t^{2} \end{matrix}]$ $e^{0.1 A} \approx$ $[\begin{matrix} 1.25 & 0.36 \\ 0.24 & 1.25 \end{matrix}]$

Exercise 3.8.105 §3.8 source ↗

For any positive integer $n$ ⁠, find a formula (or a recipe) for $A^{n}$ for the following matrices:

a) $[\begin{matrix} 7 & 4 \\ - 5 & - 2 \end{matrix}]$

b) $[\begin{matrix} - 3 & 4 \\ - 6 & 7 \end{matrix}]$

c) $[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$

Answer

a) $[\begin{matrix} 5 (3^{n}) - 2^{n + 2} & 4 (3^{n}) - 2^{n + 2} \\ 5 (2^{n}) - 5 (3^{n}) & 5 (2^{n}) - 4 (3^{n}) \end{matrix}]$ b) $[\begin{matrix} 3 - 2 (3^{n}) & 2 (3^{n}) - 2 \\ 3 - 3^{n + 1} & 3^{n + 1} - 2 \end{matrix}]$
c) $[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ if $n$ is even, and $[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$ if $n$ is odd.

Exercise 3.9.101 §3.9 source ↗

Find a particular solution to $x^{'} = 5 x + 4 y + t$ ⁠, $y^{'} = x + 8 y - t$

a) using integrating factor method,

b) using eigenvector decomposition,

c) using undetermined coefficients.

Answer

The general solution is (particular solutions should agree with one of these):
$x (t) =$ $C_{1} e^{9 t} +$ $4 C_{2} e^{4 t} -$ $^{t} /_{3} -$ $^{5} /_{54}$ ⁠, $y (t) =$ $C_{1} e^{9 t} -$ $C_{2} e^{4 t} +$ $^{t} /_{6} +$ $^{7} /_{216}$

Exercise 3.9.102 §3.9 source ↗

Find a particular solution to $x^{'} = y + e^{t}$ ⁠, $y^{'} = x + e^{t}$

a) using integrating factor method,

b) using eigenvector decomposition,

c) using undetermined coefficients.

Answer

The general solution is (particular solutions should agree with one of these):
$x (t) = C_{1} e^{t} + C_{2} e^{- t} + t e^{t}$ ⁠, $y (t) = C_{1} e^{t} - C_{2} e^{- t} + t e^{t}$

Exercise 3.9.103 §3.9 source ↗

Solve $x_{1}^{'} = x_{2} + t$ ⁠, $x_{2}^{'} = x_{1} + t$ with initial conditions $x_{1} (0) = 1$ ⁠, $x_{2} (0) = 2$ using eigenvector decomposition.

Answer

$𝐱 =$ $[\begin{matrix} 1 \\ 1 \end{matrix}] (\frac{5}{2} e^{t} - t - 1) +$ $[\begin{matrix} 1 \\ - 1 \end{matrix}] \frac{- 1}{2} e^{- t}$

Exercise 3.9.104 §3.9 source ↗

Solve $x_{1}^{''} = - 3 x_{1} + x_{2} + t$ ⁠, $x_{2}^{''} = 9 x_{1} + 5 x_{2} + \cos (t)$ with initial conditions $x_{1} (0) = 0$ ⁠, $x_{2} (0) = 0$ ⁠, $x_{1}^{'} (0) = 0$ ⁠, $x_{2}^{'} (0) = 0$ using eigenvector decomposition.

Answer

$𝐱 =$ $[\begin{matrix} 1 \\ 9 \end{matrix}] ((\frac{1}{140} +$ $\frac{1}{120 \sqrt{6}}) e^{\sqrt{6} t} +$ $(\frac{1}{140} -$ $\frac{1}{120 \sqrt{6}}) e^{- \sqrt{6} t} -$ $\frac{t}{60} -$ $\frac{\cos (t)}{70})$
$+$ $[\begin{matrix} 1 \\ - 1 \end{matrix}] (\frac{- 9}{80} \sin (2 t) +$ $\frac{1}{30} \cos (2 t) +$ $\frac{9 t}{40} -$ $\frac{\cos (t)}{30})$

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Exercises and answers are adapted from Notes on Diffy Qs: Differential Equations for Engineers by Jiří Lebl, which is free to read and download, and are used under the CC BY-NC-SA 4.0 licence. They are taken from the book’s LaTeX source and set for the web, so they may differ slightly from the printed page. This adaptation is released under the same licence and is neither affiliated with nor endorsed by Lebl or Jiří Lebl.