# MAT-285 Objectives

## MAT-285  Differential Equations

### Unit 1

• Identify differential equations as ordinary/partial, linear/nonlinear and by order.
• Define the solution of an ordinary differential equation (ODE).
• Verify the solution of an ODE.
• Identify explicit and implicit solutions to an ODE.
• Define an n-parameter family of solutions to an ODE.
• Define particular and singular solutions to an ODE.
• Define an integral curve. • Define initial value problems (IVP).
• Apply Existence of a unique Solution-Theorem for first order ODE.
• Find differential equations that model the following problems; radioactive decay, Newton’s law of cooling, tank mixing and draining, series circuits.

### Unit 2

• Use computer software to obtain a direction field.
• Use direction fields to sketch a solution to an IVP.
• Identify autonomous and non-autonomous ODEs.
• Find critical, equilibrium and stationary points to an autonomous ODE.
• Identify asymptotically stable, unstable and semi-stable critical points.
• Find phase portraits to an autonomous ODE.
• Solve a first order separable ODE. • Solve a first order linear ODE.
• Define homogeneous and non-homogeneous linear ODEs.
• Utilize the error function as a solution to a first order linear ODE.
• Use computer software to evaluate the error function.
• Solve exact ODEs. • Use integrating factors to produce exact ODEs
• Solve homogeneous ODE by substitution. • Solve a Bernoulli Differential equation.
• Use Euler’s method to approximate the solution of an IVP.
• Use computer software to facilitate computation of Euler’s method.
• Calculate absolute and relative error.

### Unit 3

• Solve linear models for the following problems; bacterial growth, half-life of a chemical substance, Newton’s law of cooling, mixture and series circuit problems.
• Solve variations of the logistic equation.
• Find a system of differential equations that model predator-prey, series circuit and mixing applications.

### Unit 4

• Apply the Existence of a Unique Solution Theorem for an nth order linear IVP.
• Contrast Initial Value Problems (IVP) with Boundary Value Problems (BVP).
• Define a system of linearly independent functions on an interval.
• Understand the form of the general solution to a linear ODE.
• Use reduction of order techniques to solve second order linear homogeneous ODEs.
• Solve linear homogeneous ODEs with constant coefficients.
• Apply the method of undetermined coefficients to find a particular solution to a non-homogeneous ODE.
• Use the method undetermined coefficients-annihilator approach to find a particular solution to a non-homogeneous ODE.
• Use variation of parameter technique to solve second order linear ODEs.
• Solve Cauchy-Euler equations by variation of parameter and undetermined Coefficient techniques.
• Use computer software to solve linear ODEs.
• Solve systems of linear ODEs by elimination.
• Use computer software to solve a system of linear ODEs.
• Solve non-linear ODEs with either independent or independent variable missing.
• Find the Taylor series solution to a nonlinear ODE.

### Unit 5

• • Solve higher order linear applications in spring/mass systems and series circuit analogue.
• Solve applied non-linear models.

### Unit 6

• Apply the Existence of a Power Series Solution Theorem for second order linear ODEs.
• Solve second order linear ODEs by Power Series. • Apply Frobenius’ theorem to second order linear ODEs.
• Solve applications of Bessel’s and Legendre’s equations.

### Unit 7

• Apply the definition of Laplace transform.
• Apply inverse Laplace transforms.
• Solve IVP and BVP using Laplace transforms.
• Use derivatives of transforms to solve ODEs.
• Use convolution theorem to solve ODEs.
• Solve Volterra integral equations using Laplace transforms.
• Use computer software to compute Laplace and inverse Laplace transforms.
• Find the Laplace transform of periodic functions. • Find the Laplace transform of the Dirac Delta function.
• Use Laplace transforms to solve a system of ODEs.

### Unit 8

• Use computer software to review basic operations of linear systems.
• Apply Existence of a Unique Solution theorem for a first order linear system (FOLS).
• Use the Wronskian to test independence of solution vectors to a FOLS.
• Understand the form of the general solution to a FOLS.
• Use computer software to calculate eigenvalues.
• Use eigenvalues to solve a FOLS.
• Use method of undetermined coefficients to solve a FOLS.
• Use method of variation of parameters to solve a FOLS.