Content check — Differential Equations (Lebl), Chapter 3: Higher-order linear ODEs

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

Chapter 3: Higher-order linear ODEs

31 problems · 0 figures

Exercise 2.1.101 §2.1 source ↗

Are $\sin (x)$ and $e^{x}$ linearly independent? Justify.

Answer

Yes. To justify try to find a constant $A$ such that $\sin (x) = A e^{x}$ for all $x$ ⁠.

Exercise 2.1.102 §2.1 source ↗

Are $e^{x}$ and $e^{x + 2}$ linearly independent? Justify.

Answer

No. $e^{x + 2} = e^{2} e^{x}$ ⁠.

Exercise 2.1.103 §2.1 source ↗

Guess a solution to $y^{''} + y^{'} + y = 5$ ⁠.

Answer

$y = 5$

Exercise 2.1.104 §2.1 source ↗

Find the general solution to $x y^{''} + y^{'} = 0$ ⁠. Hint: It is a first-order ODE in $y^{'}$ ⁠.

Answer

$y = C_{1} \ln (x) + C_{2}$

Exercise 2.1.105 §2.1 source ↗

Write down an equation (guess) for which we have the solutions $e^{x}$ and $e^{2 x}$ ⁠. Hint: Try an equation of the form $y^{''} + A y^{'} + B y = 0$ for constants $A$ and $B$ ⁠, plug in both $e^{x}$ and $e^{2 x}$ and solve for $A$ and $B$ ⁠.

Answer

$y^{''} - 3 y^{'} + 2 y = 0$

Exercise 2.2.101 §2.2 source ↗

Find the general solution to $y^{''} + 4 y^{'} + 2 y = 0$ ⁠.

Answer

$y = C_{1} e^{(- 2 + \sqrt{2}) x} + C_{2} e^{(- 2 - \sqrt{2}) x}$

Exercise 2.2.102 §2.2 source ↗

Find the general solution to $y^{''} - 6 y^{'} + 9 y = 0$ ⁠.

Answer

$y = C_{1} e^{3 x} + C_{2} x e^{3 x}$

Exercise 2.2.103 §2.2 source ↗

Find the solution to $2 y^{''} + y^{'} + y = 0$ ⁠, $y (0) = 1$ ⁠, $y^{'} (0) = - 2$ ⁠.

Answer

$y =$ $e^{- x / 4} \cos ((^{\sqrt{7}} /_{4}) x) -$ $\sqrt{7} e^{- x / 4} \sin ((^{\sqrt{7}} /_{4}) x)$

Exercise 2.2.104 §2.2 source ↗

Find the solution to $2 y^{''} + y^{'} - 3 y = 0$ ⁠, $y (0) = a$ ⁠, $y^{'} (0) = b$ ⁠.

Answer

$y =$ $\frac{2 (a - b)}{5} e^{- 3 x / 2} +$ $\frac{3 a + 2 b}{5} e^{x}$

Exercise 2.2.105 §2.2 source ↗

Find the solution to $z^{''} (t) = - 2 z^{'} (t) - 2 z (t)$ ⁠, $z (0) = 2$ ⁠, $z^{'} (0) = - 2$ ⁠.

Answer

$z (t) = 2 e^{- t} \cos (t)$

Exercise 2.2.106 §2.2 source ↗

Find the solution to $y^{''} - (α + β) y^{'} + α β y = 0$ ⁠, $y (0) = a$ ⁠, $y^{'} (0) = b$ ⁠, where $α$ ⁠, $β$ ⁠, $a$ ⁠, and $b$ are real numbers, and $α \neq β$ ⁠.

Answer

$y = \frac{a β - b}{β - α} e^{α x} + \frac{b - a α}{β - α} e^{β x}$

Exercise 2.2.107 §2.2 source ↗

Construct an equation such that $y = C_{1} e^{3 x} + C_{2} e^{- 2 x}$ is the general solution.

Answer

$y^{''} - y^{'} - 6 y = 0$

Exercise 2.3.101 §2.3 source ↗

Find the general solution of $y^{(5)} - y^{(4)} = 0$ ⁠.

Answer

$y = C_{1} e^{x} + C_{2} x^{3} + C_{3} x^{2} + C_{4} x + C_{5}$

Exercise 2.3.102 §2.3 source ↗

Suppose that the characteristic equation of a third-order differential equation has roots $\pm 2 i$ and 3.

a) What is the characteristic equation?

b) Find the corresponding differential equation.

c) Find the general solution.

Answer

a) $r^{3} - 3 r^{2} + 4 r - 12 = 0$ b) $y^{'''} - 3 y^{''} + 4 y^{'} - 12 y = 0$ c) $y =$ $C_{1} e^{3 x} +$ $C_{2} \sin (2 x) +$ $C_{3} \cos (2 x)$

Exercise 2.3.103 §2.3 source ↗

Solve $1001 y^{'''} +$ $3.2 y^{''} +$ $π y^{'} -$ $\sqrt{4} y =$ $0$ ⁠, $y (0) = 0$ ⁠, $y^{'} (0) = 0$ ⁠, $y^{''} (0) = 0$ ⁠.

Answer

$y = 0$

Exercise 2.3.104 §2.3 source ↗

Are $e^{x}$ ⁠, $e^{x + 1}$ ⁠, $e^{2 x}$ ⁠, $\sin (x)$ linearly independent? If so, show it, if not find a linear combination that gives $0$ ⁠.

Answer

No. $e^{1} e^{x} - e^{x + 1} = 0$ ⁠.

Exercise 2.3.105 §2.3 source ↗

Are $\sin (x)$ ⁠, $x$ ⁠, $x \sin (x)$ linearly independent? If so, show it, if not find a linear combination that gives $0$ ⁠.

Answer

Yes. (Hint: First note that $\sin (x)$ is bounded. Then note that $x$ and $x \sin (x)$ cannot be multiples of each other.)

Exercise 2.3.106 §2.3 source ↗

Find an equation such that $y = \cos (x)$ ⁠, $y = \sin (x)$ ⁠, $y = e^{x}$ are solutions.

Answer

$y^{'''} - y^{''} + y^{'} - y = 0$

Exercise 2.4.101 §2.4 source ↗

A mass of $2$ kilograms is on a spring with spring constant $k$ newtons per meter with no damping. Suppose the system is at rest and at time $t = 0$ the mass is kicked and starts traveling at 2 meters per second. How large does $k$ have to be so that the mass does not go further than 3 meters from the rest position?

Answer

$k =^{8} /_{9} \frac{N}{m}$ (and larger)

Exercise 2.4.102 §2.4 source ↗

Suppose we have an RLC circuit with a resistor of 100 milliohms (0.1 ohms), inductor of inductance of 50 millihenries (0.05 henries), and a capacitor of 5 farads, with constant voltage.

a) Set up the ODE equation for the current $I$ ⁠.

b) Find the general solution.

c) Solve for $I (0) = 10$ and $I^{'} (0) = 0$ ⁠.

Answer

a) $0.05 I^{''} + 0.1 I^{'} + (^{1} /_{5}) I = 0$ b) $I = C e^{- t} \cos (\sqrt{3} t - γ)$ c) $I =$ $10 e^{- t} \cos (\sqrt{3} t) +$ $\frac{10}{\sqrt{3}} e^{- t} \sin (\sqrt{3} t)$

Exercise 2.4.103 §2.4 source ↗

A $5000 kg$ railcar hits a bumper (a spring) at $1 \frac{m}{s}$ , and the spring compresses by $0.1 m$ . Assume no damping.

a) Find $k$ ⁠.

b) How far does the spring compress when a $10000 kg$ railcar hits the spring at the same speed?

c) If the spring would break if it compresses further than $0.3 m$ , what is the maximum mass of a railcar that can hit it at $1 \frac{m}{s}$ ?

d) What is the maximum mass of a railcar that can hit the spring without it breaking at $2 \frac{m}{s}$ ?

Answer

a) $k = 500000 \frac{N}{m}$ b) $\frac{1}{5 \sqrt{2}} \approx 0.141 m$ c) $45000 kg$ d) $11250 kg$

Exercise 2.4.104 §2.4 source ↗

A mass of $m$ $kg$ is on a spring with $k = 3 \frac{N}{m}$ and $c = 2 \frac{Ns}{m}$ ⁠. Find the mass $m_{0}$ for which there is critical damping. If $m < m_{0}$ ⁠, does the system oscillate or not, that is, is it underdamped or overdamped?

Answer

$m_{0} =^{1} /_{3} kg$ ⁠. If $m < m_{0}$ ⁠, then the system is overdamped and will not oscillate.

Exercise 2.5.101 §2.5 source ↗

Find a particular solution to $y^{''} - y^{'} + y = 2 \sin (3 x)$ ⁠.

Answer

$y =$ $\frac{- 16 \sin (3 x) + 6 \cos (3 x)}{73}$

Exercise 2.5.102 §2.5 source ↗

a) Find a particular solution to $y^{''} + 2 y = e^{x} + x^{3}$ ⁠.

b) Find the general solution.

Answer

a) $y = \frac{2 e^{x} + 3 x^{3} - 9 x}{6}$ b) $y =$ $C_{1} \cos (\sqrt{2} x) +$ $C_{2} \sin (\sqrt{2} x) +$ $\frac{2 e^{x} + 3 x^{3} - 9 x}{6}$

Exercise 2.5.103 §2.5 source ↗

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

Answer

$y (x) = x^{2} - 4 x + 6 + e^{- x} (x - 5)$

Exercise 2.5.104 §2.5 source ↗

Use variation of parameters to find a particular solution of $y^{''} - y = \frac{1}{e^{x} + e^{- x}}$ ⁠.

Answer

$y =$ $\frac{2 x e^{x} - (e^{x} + e^{- x}) \ln (e^{2 x} + 1)}{4}$

Exercise 2.5.105 §2.5 source ↗

For an arbitrary constant $c$ find the general solution to $y^{''} - 2 y = \sin (x + c)$ ⁠.

Answer

$y =$ $\frac{- \sin (x + c)}{3} +$ $C_{1} e^{\sqrt{2} x} +$ $C_{2} e^{- \sqrt{2} x}$

Exercise 2.5.106 §2.5 source ↗

Undetermined coefficients can sometimes be used to guess a particular solution to other equations than those with constant coefficients. Find a polynomial $y (x)$ that solves $y^{'} + x y = x^{3} + 2 x^{2} + 5 x + 2$ ⁠.

Note: Not every right-hand side will allow a polynomial solution, for example, $y^{'} + x y = 1$ does not, but a technique based on undetermined coefficients does work, see Chapter 7.

Answer

$y = x^{2} + 2 x + 3$

Exercise 2.6.101 §2.6 source ↗

A mass of $4 kg$ on a spring with $k = 4 \frac{N}{m}$ and a damping constant $c = 1 \frac{Ns}{m}$ ⁠. Suppose that $F_{0} = 2 N$ ⁠. Using the forcing function $F_{0} \cos (ω t)$ ⁠, find the $ω$ that causes practical resonance and find the practical resonance amplitude.

Answer

$ω = \frac{\sqrt{31}}{4 \sqrt{2}} \approx 0.984 \frac{rad}{s}$ $C (ω) = \frac{16}{3 \sqrt{7}} \approx 2.016 m$

Exercise 2.6.102 §2.6 source ↗

Derive a formula for $x_{s p}$ for $m x^{''} +$ $c x^{'} +$ $k x =$ $F_{0} \cos (ω t) +$ $A$ ⁠, where $A$ is some constant. Assume $c > 0$ ⁠.

Answer

$x_{s p} =$ $\frac{(ω_{0}^{2} - ω^{2}) F_{0}}{m {(2 ω p)}^{2} + m {(ω_{0}^{2} - ω^{2})}^{2}} \cos (ω t) +$ $\frac{2 ω p F_{0}}{m {(2 ω p)}^{2} + m {(ω_{0}^{2} - ω^{2})}^{2}} \sin (ω t) +$ $\frac{A}{k}$ ⁠, where $p = \frac{c}{2 m}$ and $ω_{0} = \sqrt{\frac{k}{m}}$ ⁠.

Exercise 2.6.103 §2.6 source ↗

Suppose there is no damping in a mass and spring system with $m = 5 kg$ ⁠, $k = 20 \frac{N}{m}$ ⁠, and $F_{0} = 5 N$ ⁠. Suppose $ω$ is chosen to be precisely the resonance frequency.

a) Find $ω$ ⁠.

b) Find the amplitude of the oscillations at time $t = 100$ seconds, given the system is at rest at $t = 0$ ⁠.

Answer

a) $ω = 2 \frac{rad}{s}$ b) $25 m$

Loading chapter 4…

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.