MAT-285 Differential Equations
- 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.
- 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.
- 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.
- 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.
- • Solve higher order linear applications in spring/mass systems and series circuit analogue.
- Solve applied non-linear models.
- 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.
- 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.
- 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.