MAP 2302 – INTROD TO  DIFF EQUATIONS FLORIDA INT'L UNIV.
Review for Test #2   (Sept. 9, 2008)        FALL  2008

REMEMBER TO BRING AN 8’’x11” BLUE EXAM BOOKLET FOR THE TEST

MAIN PROBLEM SOLVING TECHNIQUES:

1. How to find the Wronskian W(f₁,  f₂,  . . .  ,  f_n) of a set of functions {f₁,  f₂,  . . .  ,  f_n}.

2. How to check if a set of solutions of a linear homogeneous ODE is linearly independent.

3    Reducing the order of a linear ODE by putting y = v.f(x) where f(x) = a known solution
       4. Finding a second solution of a linear second-order ODE by putting y = v.f(x).
5. How to find the general soultion of linear homogenopus ODEs with constant coefficients
             (a) distict real roots  (b) distict complex roots (c) repeated real or complex roots.

6. How to find particular solutions of linear ODEs by the Undetermined Coefficient Method
(a) RHS is independent of complementary solution (b) RHS is not indep. of compl. sol.   

7. How to find particular solutions of linear ODEs by the Variation of Parameters Method.

8a. Solving the ODEs for a body attached to a linear spring:
(a) undamped motion,   (b) lightly damped motion,  and (c) heavily damped motion.

8b. Solving the ODEs for a body attached to a linear spring with an external force.

(a) undamped motion,   (b) lightly damped motion,  and (b) heavily damped motion

              MAIN DEFINITIONS:

        A linearly dependent set of function,  A linearly independent set of functions,  The Wronskian
        of a set of functions,  Homogeneous and non-homogeneous linear ODEs,  Auxiliary Equation,
        Multiplicity of a root of the auxiliary equation, Complementary solution, Particular solution,
        Hooke's law for linear springs,  Critically damped systems, The Resonance phenomenon.

             MAIN THEOREMS:

1. Linear independence of n solutions of an n-th order linear homogenous ODE when
             W(f₁,  f₂,  . . .  ,  f_n)  is non-zero in an interval [a,b]  (Theorem 4.4).

2. Reduction of order theorem for lin. ODEs when one non-trivial solution is given (Thm. 4.6)

3    Second-order case of  #2 with f(x) being a known non-trivial solution)  (Theorem 4.7)
             y(x) = c₁. f(x) + c₂.v(x).f(x)  where v(x) = integral of  [e^{- integral {a1(x)/a0(x)}dx} .{ f(x)}^-2]dx
       4. The D-Method for solving linear homogeneous ODEs with constant coefficients
             (a) distinct real roots  (b) distinct complex roots (c) repeated real or complex roots.

5. Finding a particular solution y_p of linear ODEs by the Undetermined Coefficient Method
(a) RHS is independent of complementary solution (b) RHS is not indep. of compl. sol.

       6. Variation of Parameters Method for finding a particular sol. from the compl. sol. y₁ and y₂ :
y_p =  v_1. y₁ + v₂ .y₂   where F(x) = RHS and  v₁ & v₂ are obtained by solving the ODEs
             (v₁)'  =  - F(x) .y₂  / [a₀(x). W(y₁,y₂)]     and    (v₂)'  =  F(x) .y₁  /  [a₀(x). W(y₁,y₂)].