## MTH 240 Section Objectives

##### Section 1.1
1. Recognize a differential equation, determine which variables are independent/dependent, and verify a solution.
2. Classify a differential equation as ordinary or partial.
3. Determine the order of a differential equation.
4. Determine whether an ordinary differential equation is linear or nonlinear.
5. Classify a solution as implicit or explicit.
6. Write a differential equation that models a problem situation.
##### Section 1.2
1. Use integration to solve an equation of the form $dy/dx=f(x)$.
##### Section 1.3
1. Construct the slope field for a differential equation.
2. Use a slope field to approximate a solution curve.
3. Use Euler's method (and other numerical methods) to approximate the solution of an initial value problem.
4. Use initial value problem existence and uniqueness theorems.
##### Section 1.4
1. Solve 1st-order separable equations.
2. Solve application problems involving separable equations, especially those involving exponential growth/decay and Newton's law of cooling.
##### Section 1.5
1. Solve 1st-order linear equations.
2. Solve application problems involving linear equations, especially those involving mixing.
##### Section 1.6
1. Use basic substitutions to solve differential equations, including Bernoulli equations, homogeneous equations, and 2nd-order equations reducible to 1st order.
2. Determine whether a differential equation is exact.
3. Solve exact differential equations.
##### Sections 2.1-2.3
1. Know and apply existence and uniqueness theorems for initial value problems associated with general linear ODE's.
2. Know and use the idea that $n$th-order, homogeneous, linear ODE's have $n$ linearly independent solutions.
3. Find and apply the Wronskian.
4. Use solutions to form a general solution for a homogeneous linear ODE.
5. Use the superposition principle.
6. Use solutions to form a general solution for a nonhomogeneous linear ODE.
7. Solve 2nd-order, constant-coefficient, homogeneous, linear ODE's.
##### Section 2.4
1. Find the equation of motion for a mass in a free, damped or undamped, mass-spring system.
2. Write the equation of motion in terms of a single sine or cosine with a phase shift.
3. Analyze the equation of motion of a mass in a mass-spring system.
##### Section 2.5
1. Use the method of undetermined coefficients to solve 2nd-order, linear, constant-coefficient, nonhomogeneous equations.
2. Use variation of parameters to solve 2nd-order, linear, nonhomogeneous equations.
##### Section 2.6
1. Find the equation of motion of a forced, undamped mass-spring system, and explain the concept of pure resonance.
2. Determine the resonance frequency of an undamped mass-spring system.
3. Find the equation of motion of a forced, damped mass-spring system, and explain the concept of practical resonance.
4. Determine the resonance frequency of a damped mass-spring system.
##### Section 3.1
1. Determine where a function is analytic.
2. Use a power series centered at an ordinary point to solve a 1st-order ODE.
##### Section 3.2
1. Use a power series centered at an ordinary point to solve a 2nd-order ODE.
##### Section 4.1
1. Define the Laplace transform of a function, and use the definition to determine a transform.
2. Use a table to determine the Laplace transform of a function.
3. Use the linearity property of the Laplace transform.
4. Know what it means for a function to be of exponential order, and use the existence/uniqueness theorems for Laplace transforms.
5. Use a table to determine the inverse Laplace transform of a function.
##### Section 4.2
1. Use Laplace transform methods to solve initial value problems.
##### Section 4.3
1. Use translation to determine Laplace transforms and inverse transforms.
2. Use partial fractions to determine inverse Laplace transforms.
##### Section 4.4
1. Use properties of Laplace transforms to find transforms and inverse transforms.
2. Use Laplace transform methods to solve equations whose coefficients are not constants.
##### Section 8.1
1. Determine the Fourier series of a function of period $2\pi$.
##### Section 8.2
1. Determine the Fourier series of a function of period $2L$.
2. Determine the convergence properties of a Fourier series.
##### Section 8.3
1. Determine the Fourier sine or cosine series of a function.
##### Section 8.5
1. Use separation of variables to solve the heat equation with Dirichlet or Neumann boundary conditions.

Last updated December 7, 2020

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