Math 250.007

Homework assignments

Homework will be due Wednesdays, in class.

Click the item to unroll the problem list.

HW 1: Due Wednesday, January 19

HW 2: Due Wednesday, January 26

Sunday, noon - updated. Tuesday - typo fixed in Part B, #1

If you started on section 2.1, save them for the future.

HW 3: Due Wednesday, February 2

Updated Sunday morning, final form.

  1. Consider the autonomous ODE below in which \(\alpha\) is a constant (real number) \(y' = \alpha + cos(y)\).
    1. For what value(s) of \(\alpha\), if any, would there be no critical points (equilibrium/constant solutions)?
    2. For what value(s) of \(\alpha\), if any, would all critical points (equilibrium/constant solutions) be semi-stable?
    3. Suppose \(\alpha = 0\) and consider the specific solution satisfying the initial condition \(y(0) = y_0\).
      Determine a value of \(y_0\) for which the solution would be increasing.
  2. Based on the plot below, find all the steady state solutions of the first-order autonomous ordinary differential equation \(df/dx = H( f )\), and determine the local stability of each. For those steady-states that are stable, determine the set of initial conditions that converge to that steady-state. You may assume $H(f) > 0 for all \(f < 4\) and \(H(f)< 0\) for all \(f > 7\).

HW 4: Due Wednesday, February 9

Update Monday, Feb 7. Note that we have not coverd Wronskians yet, so don't do the homework questions about them.

HW 5: Due Wednesday, February 16

Update Monday, Feb 14 to add the two problems from section 3.2.

Part A

Part B

  1. Consider the linear homogeneous equation \[t \left(t - 1\right) y'' + \left(1 - 2 t^{2}\right) y' + 2 \left(2 t - 1\right) y = 0.\]
    1. Show by direct calculation that \(y(t) = e^{2t}\) is a solution.
    2. Use reduction of order to find the general solution of this equation.

HW 6: Due Wednesday, February 23 (final)

Part A

Challenge (not graded)

Our textbook suggests solving 16-18 using an algebra program. Sympy is one open-source web program that will do this. Sympy is module for the python programming language, and works best best when you install it on your own computer. But for our purposes, Sympy live is good enough to solve inhomogeneous linear equations by educated guessing with undetermined coefficients. If you know python, you can use the following code example as a guide.

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from sympy import *
f = Function('f')
t = Symbol('t')
# Here's the ODE we want to solve.
ode = Eq(f(t).diff(t,t) + 3*f(t).diff(t), 2*t**4)
# pprint means 'pretty-print' for math equations
pprint(ode)

# Let's do an educated guess with undetermined coefficients
## First we have to create our coefficients, a0 ... a9.
a = symbols('a0:9')
## Now, we create a fifth-order polynomial for our guess
y = a[5]*t**5 + a[4]*t**4 + a[3]*t**3 + a[2]*t**2 + a[1]*t + a[0]

## Now, we plug our guess into the ODE and do any needed
## calculations like derivatives.
baseeq = ode.subs(f(t),y).doit().expand()
## Now, we let sympy solve for the undetermined coefficients a0..a5
coeffs = solve([ baseeq.subs(t,k) for k in range(6)], a[0:6])
## Then put the coefficients back into our guess to get the solution
ysol = y.subs(coeffs)
pprint(ysol)

# Lastly, check our solution ...
# If we subtract the right hand side (rhs) from the left side (lhs)
# and substitute in ysol for f(t), do the calculations and expand,
# everything should go to zero, since ysol is a solution.
assert (ode.rhs - ode.lhs).subs(f(t),ysol).doit().expand() == 0

HW 7: Postponed, Due Wednesday, March 16th

Formerly due Wednesday, March 2 (updated Monday)

Part A

Part B

  1. Suppose a mass - spring system has position function \(y(t) = 6e^{−2t} − 3te^{−2t}\) . Compute when the mass passes through the equilibrium position, when it hits its highest point, and then describe how low the mass goes after it hits its highest point.

  2. Suppose that when a mass of 1 kg is tied to a spring hanging from the ceiling, it stretches the spring 5/4 meters. Suppose there is a damping force with damping coefficient of 2 (in units of kg/sec). The mass is set in motion (at t = 0) from 2 above the equilibrium position, with an initial downward velocity of 2 meters/sec. Use g = 10 meters/sec 2 for gravitational acceleration.
    1. Find the position of the mass for all t > 0.
    2. Show that the time between a high point and the subsequent low point is 1/2 the quasi-period.
    3. How much time does it take to go from the equilibrium position to the next high or low point?
    4. When is the mass at its highest point?
  3. Suppose a mass-spring system is described by the ODE \(u'' + 9u = 0\) with initial conditions \(u(0) = −1\), and \(u'(0) = 0\).
    1. How high up and how far down does the mass go?
    2. What is the velocity of the mass when it passes through equilibrium?
    3. Will the mass move slower as it passes through equilibrium the 5th time it passes through compared to the 1st time?
    4. Show that the time between a high point and the subsequent low point is 1/2 the period.
    5. Show that the time between a high point and the subsequent passage through equilibrium is 1/4 the period. (You might need the identity: arctan(x) + arctan(1/x) = π/2.)
  4. Suppose a spring-mass system is undamped, and has a frequency of oscillation equal to \(2\pi\) (in units of Hertz = 1/s). Suppose it begins moving at t = 0 with initial conditions: \(u(0) = −1\) and \(u'(0) = 0\).

    1. When will it 1st pass through equilibrium and which direction will it be going?
    2. When will it reach its lowest point and how low will that be?

HW 8: Due Wednesday, March 16 (updated Saturday, March 12th)

Part A

Challenge (not graded)

  1. As mentioned in class, the Gamma function \[\Gamma(n) = \int_{0}^{\infty} t^{n-1} e^{-t} dt.\]
    • Show that \(\Gamma(1) = 1\).
    • Show that \(\Gamma(n) = n \Gamma(n-1)\).
    • The two results above imply that if \(n \in \{1,2,3,\ldots\}\), then \(\Gamma(n) = (n-1)!\). Use this result and a simple variable substitution trick to find \[\mathscr{L}[ t^k ] = \int_{0}^{\infty} e^{-st} t^k dt.\]
    • Note that this method also works for finding \(\mathscr{L}[ t^{5/3}\]\) or any other positive exponent in terms of the Gamma function, something we can not do with integration-by-parts alone.
  2. The shortest proofs to many calculus results involve complex numbers. Here's an example.
    • Assuming everything converges and the necessary integrals make sense, find \(\mathscr{L}[e^{ait}]\) when \(a\) is a real number.
    • Use the preceding result, the linearity of the Laplace transform, and the complex variables formula \[\sin(at) = (e^{ait} - e^{-ait})/2i\] to find the Laplace transform \(\mathscr{L}[\sin(at)]\).
    • Check your result against the formulas obtained in the textbook without using complex variables to confirm that they give the same answer.

HW 9: Due Wednesday, March 23

HW 10: Due Wednesday, March 30 (postponed to Friday, April 1)

HW 11: Due Wednesday, April 6 (updated Saturday)

Part A

Part B

  1. Solve the following initial-value problem using Laplace transforms: \[y'' -y' - 6y = e^{-t}u_4(t), \quad y(0)=0, \quad y'(0)=1.\]

  2. Solve the following initial-value problem using Laplace transforms: \[y'' + 4 y' + 5y = t u_2(t), \quad y(0)=0, \quad y'(0)=0.\]

HW 12: Due Wednesday, April 13

HW 13: Due Wednesday, April 20

Part A

Part B

  1. Consider the following first order linear system \[\frac{dy}{dt} = A y, \quad \text{where} \quad A = \begin{bmatrix} -9 & -1 \\ 4 & -9 \end{bmatrix} .\]
    1. Find the general solution of this system.
    2. Classify the type and stability of the stationary solution (0,0).
    3. If \(y(0) = [1,2]^T\), find the specific solution.
  2. Consider the following first order linear system \[\frac{dy}{dt} = A y, \quad \text{where} \quad A = \begin{bmatrix} 7 & -3 \\ 6 & 1 \end{bmatrix} .\]
    1. Find the general solution of this system.
    2. Classify the type and stability of the stationary solution (0,0).
  3. Consider the following first order linear system \[ \frac{dy}{dt} = A y, \quad \text{where} \quad A = \begin{bmatrix} -7 & -4 \\ 4 & 1 \end{bmatrix}. \]
    1. Find the general solution of this system.
    2. Classify the type and stability of the stationary solution (0,0).
  4. Consider the following first order linear system \[\frac{dy}{dt} = A y, \quad \text{where} \quad A = \begin{bmatrix} -4 & 1 \\ 0 & -4 \end{bmatrix} .\]
    1. Find the general solution of this system.
    2. Classify the type and stability of the stationary solution (0,0).
    3. If \(y(0) = [1,2]^T\), find the specific solution.

HW 14: Due Wednesday, April 27

For part b, there are many tools. One is to download the java 'pplane' from Alun Lloyd's page Another phaseplane simulator your can run in the browser is at Ariel Barton's, or the version at Geogebra