- Recognize a differential equation, determine which variables are independent/dependent, and verify a solution.
- Classify a differential equation as ordinary or partial.
- Determine the order of a differential equation.
- Determine whether an ordinary differential equation is linear or nonlinear.
- Classify a solution as implicit or explicit.
- Write a differential equation that models a problem situation.

- Use integration to solve an equation of the form .

- Construct the slope field for a differential equation.
- Use a slope field to approximate a solution curve.
- Use Euler's method (and other numerical methods) to approximate the solution of an initial value problem.
- Use initial value problem existence and uniqueness theorems.

- Solve 1st-order separable equations.
- Solve application problems involving separable equations, especially those involving exponential growth/decay and Newton's law of cooling.

- Solve 1st-order linear equations.
- Solve application problems involving linear equations, especially those involving mixing.

- Use basic substitutions to solve differential equations, including Bernoulli equations, homogeneous equations, and 2nd-order equations reducible to 1st order.
- Determine whether a differential equation is exact.
- Solve exact differential equations.

- Know and apply existence and uniqueness theorems for initial value problems associated with general linear ODE's.
- Know and use the idea that th-order, homogeneous, linear ODE's have linearly independent solutions.
- Find and apply the Wronskian.
- Use solutions to form a general solution for a homogeneous linear ODE.
- Use the superposition principle.
- Use solutions to form a general solution for a nonhomogeneous linear ODE.
- Solve 2nd-order, constant-coefficient, homogeneous, linear ODE's.

- Find the equation of motion for a mass in a free, damped or undamped, mass-spring system.
- Write the equation of motion in terms of a single sine or cosine with a phase shift.
- Analyze the equation of motion of a mass in a mass-spring system.

- Use the method of undetermined coefficients to solve 2nd-order, linear, constant-coefficient, nonhomogeneous equations.
- Use variation of parameters to solve 2nd-order, linear, nonhomogeneous equations.

- Find the equation of motion of a forced, undamped mass-spring system, and explain the concept of pure resonance.
- Determine the resonance frequency of an undamped mass-spring system.
- Find the equation of motion of a forced, damped mass-spring system, and explain the concept of practical resonance.
- Determine the resonance frequency of a damped mass-spring system.

- Determine where a function is analytic.
- Use a power series centered at an ordinary point to solve a 1st-order ODE.

- Use a power series centered at an ordinary point to solve a 2nd-order ODE.

- Define the Laplace transform of a function, and use the definition to determine a transform.
- Use a table to determine the Laplace transform of a function.
- Use the linearity property of the Laplace transform.
- Know what it means for a function to be of exponential order, and use the existence/uniqueness theorems for Laplace transforms.
- Use a table to determine the inverse Laplace transform of a function.

- Use Laplace transform methods to solve initial value problems.

- Use translation to determine Laplace transforms and inverse transforms.
- Use partial fractions to determine inverse Laplace transforms.

- Use properties of Laplace transforms to find transforms and inverse transforms.
- Use Laplace transform methods to solve equations whose coefficients are not constants.

- Determine the Fourier series of a function of period .

- Determine the Fourier series of a function of period .
- Determine the convergence properties of a Fourier series.

- Determine the Fourier sine or cosine series of a function.

- Use separation of variables to solve the heat equation with Dirichlet or Neumann boundary conditions.

*Last updated December 7, 2020*