2.E: Higher order linear ODEs (Exercises)

These are homework exercises to accompany Libl's "Differential Equations for Engineering" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence.

2.1: Second order linear ODEs

Exercise $\PageIndex{2.1.1}$

Show that $y = e^x$ and $y = e^{2x}$ are linearly independent.

Exercise $\PageIndex{2.1.2}$

Take $y'' + 5y = 10x + 5$. Find (guess!) a solution.

Exercise $\PageIndex{2.1.3}$

Prove the superposition principle for nonhomogeneous equations. Suppose that $y_1$ is a solution to $Ly_1 = f(x)$ and $y_2$ is a solution to $Ly_2 = g(x)$ (same linear operator $L$). Show that $y = y_1 + y_2$ solves $Ly = f(x) + g(x)$.

Exercise $\PageIndex{2.1.4}$

For the equation $x^2y'' - xy' = 0$, find two solutions, show that they are linearly independent and find the general solution. Hint: Try $y = x'$.

Equations of the form $ax^2y'' + bxy' + cy = 0$ are called Euler’s equations or Cauchy-Euler equations. They are solved by trying $y = x^r$ and solving for $r$ (we can assume that $x \geqslant 0$ for simplicity).

Exercise $\PageIndex{2.1.5}$

Suppose that ${(b - a)}^2 - 4ac > 0$.

1. Find a formula for the general solution of $ax^2y'' + bxy' + cy = 0$. Hint: Try $y = x^r$ and find a formula for $r$.
2. What happens when ${(b - a)}^2 -4ac = 0$ or ${(b - a)}^2 - 4ac < 0$?

We will revisit the case when $(b-a)^{2}-4ac<0$ later.

Exercise $\PageIndex{2.1.6}$

Same equation as in Exercise $\PageIndex{2.1.5}$. Suppose ${(b - a)}^2 - 4ac = 0$. Find a formula for the general solution of $ax^2y'' + bxy' + cy = 0$. Hint: Try $y = x^r \ln x$ for the second solution.

Exercise $\PageIndex{2.1.7}$: reduction of order

Suppose $y_1$ is a solution to $y'' + p(x)y' + q(x)y = 0$. Show that

$$y_2 (x) = y_1 (x) \int \frac { e^{- \int p(x)dx}}{{(y_1(x))}^2} dx$$

is also a solution.

Note: If you wish to come up with the formula for reduction of order yourself, start by trying $y_2 (x) = y_1(x)v(x)$. Then plug $y_2$ into the equation, use the fact that $y_1$ is a solution, substitute $w = v'$, and you have a first order linear equation in $w$. Solve for $w$ and then for $v$. When solving for $w$, make sure to include a constant of integration. Let us solve some famous equations using the method.

Exercise $\PageIndex{2.1.8}$: Chebyshev’s equation of order 1

Take $(1 - x^2)y'' - xy' + y = 0$.

1. Show that $y = x$ is a solution.
2. Use reduction of order to find a second linearly independent solution.
3. Write down the general solution.

Exercise $\PageIndex{2.1.9}$: Hermite’s equation of order 2

Take $y'' - 2xy' + 4y = 0$.

1. Show that $y = 1 - 2x^2$ is a solution.
2. Use reduction of order to find a second linearly independent solution.
3. Write down the general solution.

Exercise $\PageIndex{2.1.10}$

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

Answer

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

Exercise $\PageIndex{2.1.11}$

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

Answer

No. $e^{x+2}=e^{2}e^{x}$.

Exercise $\PageIndex{2.1.12}$

Guess a solution to $y'' + y' + y = 5$.

Answer

$y=5$

Exercise $\PageIndex{2.1.13}$

Find the general solution to $xy'' + y' = 0$. Hint: Notice that it is a first order ODE in $y'$.

Answer

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

Exercise $\PageIndex{2.1.14}$

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

Answer

$y''-3y'+2y=0$

2.2: Constant coefficient second order linear ODEs

Exercise $\PageIndex{2.2.1}$

Find the general solution of $2y'' + 2y' - 4y = 0$.

Exercise $\PageIndex{2.2.2}$

Find the general solution of $y'' + 9y' - 10y = 0$.

Exercise $\PageIndex{2.2.3}$

Solve $y'' - 8y' + 16y = 0$ for $y(0) = 2, y'(0) = 0$.

Exercise $\PageIndex{2.2.4}$

Solve $y'' + 9y' = 0$ for $y(0) = 1, y'(0) = 1$.

Exercise $\PageIndex{2.2.5}$

Find the general solution of $2y'' + 50y = 0$.

Exercise $\PageIndex{2.2.6}$

Find the general solution of $y'' + 6y' + 13y = 0$.

Exercise $\PageIndex{2.2.7}$

Find the general solution of $y'' = 0$ using the methods of this section.

Exercise $\PageIndex{2.2.8}$

The method of this section applies to equations of other orders than two. We will see higher orders later. Try to solve the first order equation $2y' + 3y = 0$ using the methods of this section.

Exercise $\PageIndex{2.2.9}$

Let us revisit Euler’s equations of Exercise $\PageIndex{1}$. Suppose now that ${(b - a)}^2 - 4ac < 0$. Find a formula for the general solution of $ax^2y'' + bxy' + cy = 0$. Hint: Note that $x^r = e^{r \ln x}$.

Exercise $\PageIndex{2.2.10}$

Find the solution to $y''-(2\alpha )y'+\alpha^{2}y=0$, $y(0)=a$, $y'(0)=b$, where $\alpha$, $a$, and $b$ are real numbers.

Exercise $\PageIndex{2.2.11}$

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

Exercise $\PageIndex{2.2.12}$

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

Answer

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

Exercise $\PageIndex{2.2.13}$

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

Answer

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

Exercise $\PageIndex{2.2.14}$

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

Answer

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

Exercise $\PageIndex{2.2.15}$

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

Answer

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

Exercise $\PageIndex{2.2.16}$

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

Answer

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

Exercise $\PageIndex{2.2.17}$

Find the solution to $y''-(\alpha +\beta )y'+\alpha\beta y=0$, $y(0)=a$, $y'(0)=b$, where $\alpha$, (\beta\), $a$, and $b$ are real numbers, and $\alpha\neq\beta$.

Answer

$y=\frac{\alpha\beta -b}{\beta -\alpha}e^{\alpha x}+\frac{b-a\alpha}{\beta -\alpha}e^{\beta x}$

Exercise $\PageIndex{2.2.18}$

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

Answer

$y''-y'-6y=0$

2.3: Higher order linear ODEs

Exercise $\PageIndex{2.3.1}$

Find the general solution for $y''' - y'' + y' - y = 0$.

Exercise $\PageIndex{2.3.2}$

Find the general solution for $y^{(4)} - 5y''' + 6y'' = 0$.

Exercise $\PageIndex{2.3.3}$

Find the general solution for $y''' + 2y'' + 2y' = 0$.

Exercise $\PageIndex{2.3.4}$

Suppose the characteristic equation for a differential equation is ${(r - 1)}^2{(r -2)}^2 = 0$.

1. Find such a differential equation.
2. Find its general solution.

Exercise $\PageIndex{2.3.5}$

Suppose that a fourth order equation has a solution $y = 2e^{4x} x \cos x$.

1. Find such an equation.
2. Find the initial conditions that the given solution satisfies.

Exercise $\PageIndex{2.3.6}$

Find the general solution for the equation of Exercise $\PageIndex{2.3.5}$.

Exercise $\PageIndex{2.3.7}$

Let $f(x) = e^x - \cos x, g(x) = e^x + \cos x$ and $h(x) = \cos x$. Are $f(x), g(x),$ and $h(x)$ and linearly independent? If so, show it, if not, find a linear combination that works.

Exercise $\PageIndex{2.3.8}$

Let $f(x) = 0, g(x) = \cos x$, and $h(x) = \sin x$. Are $f(x), g(x),$ and $h(x)$ and linearly independent? If so, show it, if not, find a linear combination that works.

Exercise $\PageIndex{2.3.9}$

Are $x, x^2, and x^4$ linearly independent? If so, show it, if not, find a linear combination that works.

Exercise $\PageIndex{2.3.10}$

Are $e^x, xe^x, and x^2e^x$ linearly independent? If so, show it, if not, find a linear combination that works.

Exercise $\PageIndex{2.3.11}$

Find an equation such that $y=xe^{-2x}\sin (3x)$ is a solution.

Exercise $\PageIndex{2.3.12}$

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 $\PageIndex{2.3.13}$

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

1. What is the characteristic equation?
2. Find the corresponding differential equation.
3. Find the general solution.

Answer

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

Exercise $\PageIndex{2.3.14}$

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

Answer

$y=0$

Exercise $\PageIndex{2.3.15}$

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

Answer

No. $e^{1}e^{x}-e^{x+1}=0$.

Exercise $\PageIndex{2.3.16}$

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

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 $\PageIndex{2.3.17}$

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

Answer

$y'''-y''+y'-y=0$

2.4: Mechanical Vibrations

Exercise $\PageIndex{2.4.1}$

Consider a mass and spring system with a mass $m = 2$, spring constant $k = 3$, and damping constant $c = 1$.

1. Set up and find the general solution of the system.
2. Is the system underdamped, overdamped or critically damped?
3. If the system is not critically damped, find a $c$ that makes the system critically damped.

Exercise $\PageIndex{2.4.2}$

Do Exercise $\PageIndex{2.4.1}$ for $m = 3, k = 12, and c = 12$.

Exercise $\PageIndex{2.4.3}$

Using the mks units (meters-kilograms-seconds), suppose you have a spring with spring constant $4 \dfrac {N}{m}$. You want to use it to weigh items. Assume no friction. You place the mass on the spring and put it in motion.

1. You count and find that the frequency is $0.8\text{ Hz}$ (cycles per second). What is the mass?
2. Find a formula for the mass $m$ given the frequency $w$ in $\text{Hz}$.

Exercise $\PageIndex{2.4.4}$

Suppose we add possible friction to Exercise $\PageIndex{2.4.3}$. Further, suppose you do not know the spring constant, but you have two reference weights $1\text{ kg}$ and $2\text{ kg}$ to calibrate your setup. You put each in motion on your spring and measure the frequency. For the $1\text{ kg}$ weight you measured $1.1\text{ Hz}$, for the $2\text{ kg}$ weight you measured $0.8\text{ Hz}$.

1. Find $k$ (spring constant) and $c$ (damping constant).
2. Find a formula for the mass in terms of the frequency in $\text{Hz}$. Note that there may be more than one possible mass for a given frequency.
3. For an unknown object you measured \(0.2\text{ Hz}\), what is the mass of the object? Suppose that you know that the mass of the unknown object is more than a kilogram.

Exercise $\PageIndex{2.4.5}$

Suppose you wish to measure the friction a mass of $0.1\text{ kg}$ experiences as it slides along a floor (you wish to find $c$). You have a spring with spring constant $k = 5 \dfrac {N}{m}$. You take the spring, you attach it to the mass and fix it to a wall. Then you pull on the spring and let the mass go. You find that the mass oscillates with frequency $1\text{ Hz}$. What is the friction?

Exercise $\PageIndex{2.4.6}$

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 to so that the mass does not go further than $3$ meters from the rest position?

Answer

$k=\frac{8}{9}$ (and larger)

Exercise $\PageIndex{2.4.7}$

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

1. Set up the ODE equation for the current $I$.
2. Find the general solution.
3. Solve for $I (0) = 10$ and $I' (0) = 0$.

Answer

1. $0.05I''+0.1I'+\left(\frac{1}{5}\right)I=0$
2. $I=Ce^{-t}\cos (\sqrt{3}t-\gamma )$
3. $I=10e^{-t}\cos (\sqrt{3} t)+\frac{10}{\sqrt{3}}e^{-t}\sin (\sqrt{3}t)$

Exercise $\PageIndex{2.4.8}$

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

1. Find $k$.
2. Find out how far does the spring compress when a $10000\text{ kg}$ railcar hits the spring at the same speed.
3. If the spring would break if it compresses further than $0.3\text{ m}$, what is the maximum mass of a railcar that can hit it at $1 \dfrac {m}{s}$?
4. What is the maximum mass of a railcar that can hit the spring without breaking at $2 \dfrac {m}{s}$?

Answer

1. $k=500000$
2. $\frac{1}{5\sqrt{2}}\approx 0.141$
3. $45000\text{ kg}$
4. $11250\text{ kg}$

Exercise $\PageIndex{2.4.9}$

A mass of $m\text{ kg}$ is on a spring with $k=3\frac{\text{N}}{\text{m}}$ and $c=2\frac{\text{Ns}}{\text{m}}$. 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

$m_{0}=\frac{1}{2}$. If $m<m_{0}$, then the system is overdamped and will not oscillate.

2.5: Nonhomogeneous Equations

Exercise $\PageIndex{2.5.1}$

Find a particular solution of $y'' - y' - 6y = e^{2x}$.

Exercise $\PageIndex{2.5.2}$

Find a particular solution of $y'' - 4y' + 4y = e^{2x}$.

Exercise $\PageIndex{2.5.3}$

Solve the initial value problem $y'' + 9y = \cos (3x) + \sin (3x)$ for $y(0) = 2, y'(0) = 1$.

Exercise $\PageIndex{2.5.4}$

Setup the form of the particular solution but do not solve for the coefficients for $y^{(4)} - 2y''' + y'' = e^x$.

Exercise $\PageIndex{2.5.5}$

Setup the form of the particular solution but do not solve for the coefficients for $y^{(4)} - 2y''' + y'' = e^x + x + \sin x$.

Exercise $\PageIndex{2.5.6}$

1. Using variation of parameters find a particular solution of $y'' - 2y' + y = e^x$.
2. Find a particular solution using undetermined coefficients.
3. Are the two solutions you found the same? What is going on? See also Exercise $\PageIndex{2.5.9}$

Exercise $\PageIndex{2.5.7}$

Find a particular solution of $y'' - 2y' + y = \sin (x^2)$. It is OK to leave the answer as a definite integral.

Exercise $\PageIndex{2.5.8}$

For an arbitrary constant $c$ find a particular solution to $y'' - y = e^{cx}$. Hint: Make sure to handle every possible real $c$.

Exercise $\PageIndex{2.5.9}$

1. Using variation of parameters find a particular solution of $y''-y=e^{x}$
2. Find a particular solution using undetermined coefficients.
3. Are the two solutions you found the same? What is going on?

Exercise $\PageIndex{2.5.10}$

Find a polynomial $P(x)$, so that $y=2x^{2}+3x+4$ solves $y''+5y'+y=P(x)$.

Exercise $\PageIndex{2.5.11}$

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

Answer

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

Exercise $\PageIndex{2.5.12}$

1. Find a particular solution to $y'' + 2y = e^x + x^3$.
2. Find the general solution.

Answer

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

Exercise $\PageIndex{2.5.13}$

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

Answer

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

Exercise $\PageIndex{2.5.14}$

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

Answer

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

Exercise $\PageIndex{2.5.15}$

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

Answer

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

Exercise $\PageIndex{2.5.16}$

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

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

Answer

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

2.6: Forced Oscillations and Resonance

Exercise $\PageIndex{2.6.1}$

Derive a formula for $x_{sp}$ if the equation is $mx'' + cx' + kx = F_0 \sin ( \omega t)$. Assume $c > 0$.

Exercise $\PageIndex{2.6.2}$

Derive a formula for $x_{sp}$ if the equation is $mx'' + cx' + kx = F_0 \cos (\omega t) + F_1 \cos (3 \omega t)$. Assume $c > 0$.

Exercise $\PageIndex{2.6.3}$

Take $mx'' + cx' + kx = F_0 \cos ( \omega t)$. Fix $m > 0$ and $k > 0$. Now think of the function $C(\omega )$. For what values of $c$ (solve in terms of $m, k,$ and $F_0$ ) will there be no practical resonance (that is, for what values of $c$ is there no maximum of $C(\omega )$ for $\omega > 0$ )?

Exercise $\PageIndex{2.6.4}$

Take $mx'' + cx' + kx = F_0 \cos (\omega t)$. Fix $c > 0$ and $k > 0$. Now think of the function $C(\omega )$. For what values of $m$ (solve in terms of $c, k,$ and $F_0$) will there be no practical resonance (that is, for what values of $m$ is there no maximum of $C(\omega )$ for $\omega > 0$ )?

Exercise $\PageIndex{2.6.5}$

Suppose a water tower in an earthquake acts as a mass-spring system. Assume that the container on top is full and the water does not move around. The container then acts as a mass and the support acts as the spring, where the induced vibrations are horizontal. Suppose that the container with water has a mass of $m = 10, 000 kg$. It takes a force of $1000$ newtons to displace the container $1$ meter. For simplicity assume no friction. When the earthquake hits the water tower is at rest (it is not moving).

Suppose that an earthquake induces an external force $F(t) = mA \omega^2 \cos (\omega t).$

1. What is the natural frequency of the water tower?
2. If $\omega$ is not the natural frequency, find a formula for the maximal amplitude of the resulting oscillations of the water container (the maximal deviation from the rest position). The motion will be a high frequency wave modulated by a low frequency wave, so simply find the constant in front of the sines.
3. Suppose $A = 1$ and an earthquake with frequency $0.5$ cycles per second comes. What is the amplitude of the oscillations? Suppose that if the water tower moves more than $1.5$ meter, the tower collapses. Will the tower collapse?

Exercise $\PageIndex{2.6.6}$

A mass of $4\text{ kg}$ on a spring with $k = 4$ and a damping constant $c = 1$. Suppose that $F_0 = 2$. Using forcing of $F_0 \cos (\omega t)$. Find the $\omega$ that causes practical resonance and find the amplitude.

Answer

$\omega=\frac{\sqrt{31}}{4\sqrt{2}}\approx 0.984\quad C(\omega )=\frac{16}{3\sqrt{7}}\approx 2.016$

Exercise $\PageIndex{2.6.7}$

Derive a formula for $x_{sp}$ for $mx'' + cx' + kx = F_0 \cos (\omega t) + A$ where $A$ is some constant. Assume $c > 0$.

Answer

$x_{sp}=\frac{(\omega_{0}^{2}-\omega^{2})F_{0}}{m(2\omega p)^{2}+m(\omega_{0}^{2}-\omega^{2})^{2}}\cos (\omega t)+\frac{2\omega pF_{0}}{m(2\omega p)^{2}+m(\omega_{0}^{2}-\omega^{2})^{2}}\sin (\omega t)+\frac{A}{k}$, where $p=\frac{c}{2m}$ and $\omega_{0}=\sqrt{\frac{k}{m}}$

Exercise $\PageIndex{2.6.8}$

Suppose there is no damping in a mass and spring system with $m = 5, k = 20,$ and $F_0 = 5$. Suppose that $\omega$ is chosen to be precisely the resonance frequency.

1. Find $\omega$.
2. Find the amplitude of the oscillations at time $t = 100$.

Answer

1. $\omega =2$
2. $25$