*Standards marked with “m” are modeling standards.*

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 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.

## Module D

Solve and apply 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.