MTH 240 Section Objectives

Steve Kifowit, Waubonsee Community College

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)$$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$$n$th-order, homogeneous, linear ODE's have $n$$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 constant-coefficient, homogeneous, linear ODE's of all orders.
8. Solve 2nd-order, homogeneous, Cauchy-Euler equations.

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.
4. Determine if a mass-spring system is undamped, underdamped, overdamped, or critically damped.

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$$2\pi$.

Section 8.2

1. Determine the Fourier series of a function of period $2L$$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 October 14, 2024

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