MAT 227 - Diff Equations with Laplace

1. Discuss differential equations and their solutions
   1. Classify a differential equation by its properties
   2. Verify a proposed solution to a differential equation
   3. Interpret a direction field for a differential equation
   4. Utilize the method of isoclines
   5. Exhibit the existence and uniqueness of a differential equation’s solution

2. Solve first-order differential equations
   1. Solve separable differential equations
   2. Assess exactness of a differential equation
   3. Convert a differential equation into an exact form
   4. Solve an exact differential equation using integration
   5. Utilize an integrating factor to solve a linear differential equation
   6. Recognize substituted forms of first order differential equations

3. Solve second-order differential equations
   1. Identify homogeneous and non-homogeneous second order differential equations
   2. Examine real and imaginary roots of a characteristic equation
   3. Solve a differential equation using the method of undetermined coefficients
   4. Construct a solution using the superposition principle
   5. Solve a non-homogeneous differential equation using variation of parameters
   6. Identify Cauchy-Euler equations

4. Adapt differential equations in mathematical models
   1. Examine applications of first order differential equations in modeling
   2. Examine applications of second order differential equations in modeling
   3. Model applications with a system of differential equations

5. Solve systems of first order differential equations
   1. Assess independence of solutions with the Wronskian
   2. Utilize the elimination method to solve systems
   3. Solve non-homogeneous systems with variation of parameters

6. Solve differential equations of order three or higher
   1. Analyze a differential equation with operators
   2. Utilize the method of annihilators to solve a higher-order differential equation
   3. Use variation of parameters on higher-order differential equations

7. Solve differential equations using Laplace transforms
   1. Define the Laplace transform as an integral
   2. Apply the Laplace transform to a differential equation
   3. Construct the inverse Laplace transform for common functions
   4. Define the convolution of two functions
   5. Solve differential equations with the Laplace transform and inverse transform
   6. Make use of the Heaviside function to solve differential equations
   7. Solve differential equations using the Dirac delta function

8. Solve differential equations using numerical methods
   1. Utilize numerical methods to solve first-order differential equations
   2. Compute solutions to second-order differential equations with numerical methods
   3. Solve systems of differential equations using numerical methods