# MA 238 Standards

Differential Equations - 2020 Spring

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.