STANDARD COMPETENCIES:

 

 I.      Identify definitions and family of curves.

 II.     Solve equations of order one.  (I, II, III)

 III.    Research and apply additional topics on equations of order one.

 IV.     Solve linear differential equations.

 V.      Solve linear equations with constant coefficients.

 VI.     Solve non-homogenous equations.  (VI, VII)

 VII.    Solve problems involving variation of parameters.

 VIII.   Solve Linear Systems of Equations.  (VIII)

 IX.     Find Power Series Solutions. (IX, X)

 X       Analyze solutions near regular singular points.

 TOPICAL OUTLINE:

 

 I.      Identify definitions and families of curves.

         A.      Give examples of differential equations.

         B.      Reproduce and explain definitions.

         C.      Examine families of solutions.

         D.      Interpret geometrically.

         E.      Isolation an equation.

         F.      Use and apply an existence theorem.

 II.     Solve equations of order one.

         A.      Use the separations of variables method to solve problems.

         B.      Solve homogenous functions.

         C.      Solve equations with homogenous coefficients.

         D.      Find exact equations.

         E.      Solve linear equation of order one.

         F.      Find the general solution of a linear equation.

 III.    Research and apply additional topics on equations of order one.

         A.      Find integrating factors by inspection.

         B.      Examine the determination of integrating factors.

         C.      Observe substitutions suggested by the equation.

         D.      Illustrate Bernoulli¿s Equation.

         E.      Determine coefficients linear in the two variables.

         F.      Analyze solutions involving non-elementary integrals.

 IV.     Solve linear differential equations.

         A.      Solve general linear equations.

         B.      Illustrate an Existence and Uniqueness Theorem.

         C.      Illustrate linear independence.

         D.      Use the Wronskian to show linear independence.

         E.      Find the general solution of a homogenous equation.

         F.      Find the general solution of a non-homogenous equation.

         G.      Use differential operators to solve problems.

         H.      Illustrate the Fundamental Laws of Operations.

         I.      Illustrate properties of Differential Operators.

 V.      Solve linear equations with constant coefficients.

         A.      Find distinct roosts using the Auxiliary Equation.

         B.      Find repeated roots using the Auxiliary Equation.

         C.      Illustrate the definition of exp z for Imaginary z.

         D.      Find imaginary roots using the Auxiliary Equation.

         E.      Examine hyperbolic functions.

 VI.     Solve non-homogenous equations: Undetermined Coefficients.

         A.      Construct a homogenous equation from a specific solution.

         B.      Solve non-homgenous equations.

         C.      Use the Method of Undetermined Coefficients to solve problems.

         D.      Solve non-homogeneous equations by Inspection.

 VII.    Solve problems involving variation of parameters.

         A.      Use Reduction of Order to obtain explicit solutions of differential equations.

         B.      Solve problems involving variation of parameters.

         C.      Find the solution of y¿ + y = f(x).

 VIII.   Solve linear systems of equations.

         A.      Solve first-order systems with constant coefficients.

         B.      Analyze the solutions of a first-order systems.

         C.      Perform matrix algebra.

         D.      Find the values for complex eigenvalues

         E.      Find the values for repeated eigenvalues

 IX.     Find power series solutions.

         A.      Solve linear equations and power series.

         B.      Analyze convergence of power series.

         C.      Analyze ordinary points and singular points.

         D.      Illustrate the validity of the solutions near an ordinary point.

         E.      Analyze the solutions near an ordinary point.

 X       Analyze solutions near regular singular points.

         A.      Identify regular singular points.

         B.      Solve the indicial equation.

         C.      Analyze the form and validity of the solutions near a regular singular point.

         D.      Analyze indicial equations with difference of nonintegral roots.

         E.      Analyze differentiation of a product of functions.

         F.      Analyze indicial equations with equal roots.

         G.      Analyze indicial equations with equal roots: An alternative.

         H.      Analyze indicial equations with difference of roots a positive integer: Nonlogarithmic case.

         I.      Analyze indicial equations with difference of roots a positive integer: Logarithmic case.