\( \newcommand{\sech}{\operatorname{sech}} \) \( \newcommand{\inverse}[1]{#1^\leftarrow} \) \( \newcommand{\<}{\langle} \) \( \newcommand{\>}{\rangle} \) \( \newcommand{\vect}{\mathbf} \) \( \newcommand{\veci}{\mathbf{\hat ı}} \) \( \newcommand{\vecj}{\mathbf{\hat ȷ}} \) \( \newcommand{\veck}{\mathbf{\hat k}} \) \( \newcommand{\curl}{\operatorname{curl}\,} \) \( \newcommand{\dv}{\operatorname{div}\,} \) \( \newcommand{\detThree}[9]{ \operatorname{det}\left( \begin{array}{c c c} #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right) } \) \( \newcommand{\detTwo}[4]{ \operatorname{det}\left( \begin{array}{c c} #1 & #2 \\ #3 & #4 \end{array} \right) } \)

MA 238 Standards

Differential Equations - 2020 Spring
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  3. 2020
  4. Spring
  5. MA 237/238
  6. MA 238 Standards

Standards marked with “m” are asssed with take-home projects.

At the end of this course, you should be able to…

Module C

Solve and apply linear constant-coefficient ODEs.

  • C1: Homogeneous first-order constant coefficient. Solve homogeneous linear constant coefficient first-order ODEs.
  • C2: Non-homogeneous first-order constant coefficient. Solve nonhomogeneous linear constant coefficient first-order ODEs.
  • C3m: Motion with linear drag. Model and analyze the vertical motion of an object with linear drag
  • C4: Homogeneous second-order constant coefficient. Solve homogeneous linear constant coefficient second-order ODEs.
  • C5: Initial value problems. Solve homogeneous linear constant coefficient second-order IVPs.
  • C6: Non-homogeneous second-order constant coefficient. Solve nonhomogeneous linear constant coefficient second-order ODEs.
  • C7m: Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP.

Module D

Solve and apply linear constant coefficient ODEs that involve discontinuous functions or distributions.

  • D1: Laplace transform. Compute the Laplace transform of a function from the definition.
  • D2: Discontinuous IVPs. Use Laplace transforms to solve IVPs involving a step function or Dirac delta distribution.
  • D3m: Non-smooth motion. Model and analyze motion involving instantaneous acceleration.
  • D4m: Mass-spring impulse. Model and analyze motion of a mass-spring system involving an impulse.

Module F

Solve and apply first-order ODEs.

  • F1: Sketching trajectories. Sketch the trajectory of the solution of a first-order ODE given its slope field.
  • F2: Separable IVPs. Find the solution to a separable IVP.
  • F3m: Motion with quadratic drag. Model and analyze the horizontal motion of an object with quadratic drag
  • F4: Autonomous ODEs. Sketch and label the phase line of an autonomous ODE, and use it to determine the long-term behavior of solutions.
  • F5: First-order linear IVPs. Find the solution to a first-order linear IVP.
  • F6: Exact ODEs. Find the implicit general solution to a first-order exact ODE.

Module S

Solve and apply systems of ODEs.

  • S1: Constant coefficient systems. Solve systems of first-order constant-coefficient IVPs.
  • S2m: Coupled mass-spring systems. Model and analyze mechanical oscillators with a system of second-order IVPs.
  • S3: Autonomous systems. Sketch and label the phase plane of an autonomous system of ODEs.
  • S4m: Interacting populations. Model and analyze two interacting populations with an autonomous system of IVPs.

Module N

Use numerical approximation methods to analyze unsolvable IVPs.

  • N1: Existence and uniqueness. Apply an existence and uniqueness theorem to a second-order linear IVP.
  • N2: Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method.
  • N3m: Programming Euler’s method. Implement Euler’s method using technology.