Outline for Math350

Chapter 2.1-2.9: First Order Differential Equations

1. Concepts & Terminologies:
Section 1.1		 Ordinary and Partial Differential Equations;
Systems of DE;
Order;
Solution;
Linear and Nonlinear DE;
Direction Fields - PlotVectorField
Linear Equations:
Section 2.1 		Separation of variables;
Initial conditions;
DSolve and direction fields;
Integrating factors;
  Homework (Due 9/8/99): 2.1:  #1, #7, #24, #27
2. Section 2.1 (cont.) Integration curves
Section 2.2 Existence and uniqueness of solutions to initial value problems.  Theorem 2.2.1
Homework (Due 9/13/99): 2.2:  #9, #10
3. Section 2.3 Separable Equations
        M(x)dx+N(y)dy = 0
  Homework 2.3:  #9, #19, #29
4. Section 2.4 Differences between linear and nonlinear equations
   Uniqueness
   Interval of definition
   Difference in the concepts of general solutions
   Implicit solutions
Graphical/Numerical Construction of Integral curve
  Homework 2.4: #4, $14, #17
5. Section 2.5 Modeling with linear equations

Basic procedure for modeling

  1. Physical laws;
  2. Gathering known information;
  3. Setting up mathematical equations;
  4. Solving the mathematical equations and identifying all parameters;
  5. Checking the solution with reality.  If the solution is inconsistent with reality, make necessary adjustments, and reconstruct the model.

Ex1. Radioactive Decay; 
Ex2. Compound Interest;
Ex3. Mixing;
Ex4. Determination of the Time of Death;
  Homework 2.5: #18, #19, #27
6. Section 2.6 Population Dynamics and Some Related Problems
  • Autonomous equations;
  • Exponential growth and Logistic growth;
  • Equilibrium solutions and critical points;
  • Stability of equilibrium solutions;
  • Critical threshold;
  • Logistic equation with a threshold;
  Homework 2.6: #3, #20, #24
7. Section 2.7 Mechanics Problems
  • Newton's law of motion: F = ma;
  • gravitational force: w(x) = mgR2/(R+x)2;
  Homework 2.7: #1, #7
8. Section 2.8 Exact Equations and Integrating Factors
             M(x,y)+N(x,y)y' = 0
  • Criterion for an equation to be exact: Theorem 2.8.1;
  • Solutions of exact equations;
  • Integrating factors;
  Homework 2.8: #13, #28
9. Section 2.9 Homogeneous Equations
             y' = F(y/x)

Transformation: v = y/x,  
i.e.
         y = x v

  Homework 2.9: #14, #16
     

Chapter 8: Numerical Methods

10. The Euler Method (Tangent Line)
Section 8.1		             y' = f(t,y), y(t0) = y0

Tangent Line Approximation:

          yn+1 = yn+ h f(tn, yn)
Homework 8.1: #3(a-b), #12(a-b)
11. Errors in Numerical Procedures
Section 8.2

Convergence
Global truncation error: En
Local truncation error: en
Round-off error: Rn
12. Improved Euler Method (Heun)
Section 8.3		          yn+1 = yn+ h [f(tn, yn) + f(tn+h, yn+ h f(tn, yn)) ]/2
Homework 8.3: #6, #12
13. Runge-Kutta Method
Section 8.4		          yn+1 = yn+ h [K1 + 2K2 + 2K3 + K4 ]/6
where
         K1 = f(tn, yn)
           K2 = f(tn + 0.5h, yn + 0.5hK1)
           K3 = f(tn + 0.5h, yn + 0.5hK2 )
           K4 = f(tn + h, yn + hK3)

Test 1

14. Review Chapter 2.1-2.9 and 8.1-8.3
15. Test 1 (Friday, 10/15/99) Note: First one-hour test.

Chapter 3: Second Order Linear Equations

16. Homogeneous Equation with Constant Coefficients
Section 3.1		 Equation:
          a y'' + b y' + c y = 0

Characteristic Equation:
          a r2 + b r + c =0 

Case of 
          b2 - 4ac > 0 

Solutions of the characteristic equation: 
          r1 , r2 are real numbers

General solution:
          y(t) = C1 Exp(r1 t) + C2 Exp(r2 t)

Initial Condition:
          y(0) = y0, y'(0) = y'0

Example: 
          For what values of b, the nonzero solutions of y'' + b y' + y = 0 goes to zero as t goes to infinity?
Homework 3.1: #1, #9, #23
17. Fundamental Solutions
Section 3.2 & 3.3
  • Equation:
              L[y] = y'' + p(t) y' + q(t) y = 0
  • Existence and Uniqueness: Theorem 3.2.1
  • Principle of Superposition:
    If f(t) and g(t) are solutions, then k1 f(t) + k2 g(t) is also a solution.
  • Wronskian of two functions:
              W(y1, y2)(t) = y1(t) y2'(t) - y1'(t) y2(t)
  • If y1, y2 are solutions, then,
             W(y1, y2)(t) = c Exp[- Integrate[p(t),t]]
    where c is a constant.
  • Linear independence of two functions f(t) and g(t)
            k1 f(t) + k2 g(t) =     0
    if and only if k1= 0 and k2 = 0.
  • If two solutions f(t) and g(t) are linear independent, they are called a set of fundamental solutions, and the general solution is given by
            y(t) = k1 f(t) + k2 g(t)
Homework 3.2: #21
3.3: #2
18. Complex Roots of the Characteristic Equation
Section 3.4		 Equation:
          a y'' + b y' + c y = 0

Characteristic Equation:
          a r2 + b r + c =0 

Case of 
          b2 - 4ac < 0 

Solutions of the characteristic equation: 
          r1, r2  are a pair of complex numbers 

         

General solution:
         

Initial Condition:
          y(0) = y0, y'(0) = y'0
Homework 3.4: #19
19. Homogeneous Equation with Constant Coefficients
Section 3.5		 Equation:
          a y'' + b y' + c y = 0

Characteristic Equation:
          a r2 + b r + c =0 

Case of 
          b2 - 4ac = 0 

Solutions of the characteristic equation: 
          r1 = r2 are real numbers

General solution:
          y(t) = (C1  + C2 t)*Exp(r1 t) = ( C1  + C2 t)*Exp(r2 t)

Initial Condition:
          y(0) = y0, y'(0) = y'0

Reduction of Order

For equation:
          y'' + p(t) y' + q(t) y = 0

if y1 is a solution, then the transform 
         y = v y1
will change the equation into simple equation
         y1v'' + (2 y1 + p y1) v'  = 0,
which is basically a linear first order equation, 
         y1u' + (2 y1 + p y1) u = 0 
where u = v' .
Homework 3.5: #2, #23
20. Non-homogeneous Equation: Method of Undetermined Coefficients
Section 3.6		 If the equation:
          y'' + p(t) y' + q(t) y = 0
has the general solution as:
         y(t) = C1 y1  + C2 y2 
and the equation:
          y'' + p(t) y' + q(t) y = g(t)
has a particular solution Y(t), then the general solution of 
          y'' + p(t) y' + q(t) y = g(t)
is 
         y(t) = C1 y1(t)  + C2 y2(t) + Y(t)
Homework 3.6: #2, #7
21. Non-homogeneous Equation: Variation of Parameters
Section 3.7		 Suppose that the equation:
          y'' + p(t) y' + q(t) y = 0
has the general solution as:
         y(t) = C1 y1  + C2 y2 

Assume that general solution of the equation:
          y'' + p(t) y' + q(t) y = g(t)
has the form
         y(t) = u1(t)y1(t)  + u2(t) y2(t) 
where u1(t) and u2(t) are functions to be determined. 
Homework 3.7: #2, #17
22. Applications: Mechanical/Electrical Vibrations
Section 3.8		 Mechanical Vibrations:
          m u'' + r u' + k u = F(t), u(0) = u0, u'(0) = u0'
Electrical Vibrations:
          L Q'' + R Q' + 1/C Q = E(t), Q(0) = Q0, Q'(0) = Q0' = I0
Homework 3.8: #5, #9
23. Applications: Mechanical/Electrical Vibrations (cont)
Section 3.9		 Mechanical Vibrations:
          m u'' + r u' + k u = F(t), u(0) = u0, u'(0) = u0'
Electrical Vibrations:
          L Q'' + R Q' + 1/C Q = E(t), Q(0) = Q0, Q'(0) = Q0' = I0

Chapter 4: Higher Order Linear Equations

24. Higher Order Linear Equations
Section 4.1		 Consider
          P0 y(n) + P1 y(n-1) + P2 y(n-2)  + ... + Pn y = G(t)
Initial Condition:
          y(t0) = y0 , y'(t0) = y0', ... y(n-1)(t0) = y0(n-1) 

Homogeneous Equations: Wronskian, fundamental set of solutions

Non-Homogeneous Equations: particular solution and general solution
Homework 4.1: #10, #12
25. Higher Order Linear Equations: Homogeneous Equations with Constant Coefficients
Section 4.2		 Consider
          a0 y(n) + a1 y(n-1) + a2 y(n-2)  + ... + an y = 0
Initial Condition:
          y(t0) = y0 , y'(t0) = y0', ... y(n-1)(t0) = y0(n-1) 
Homework 4.2: #11, #35
26. Non-homogeneous Equation: Method of Undetermined Coefficients
Section 4.3		 If the equation:
          P0 y(n) + P1 y(n-1) + P2 y(n-2)  + ... + Pn y = 0
has the general solution as:
         y(t) = C1 y1  + C2 y2 + ... + Cn yn
and the equation:
         P0 y(n) + P1 y(n-1) + P2 y(n-2)  + ... + Pn y = G(t)
has a particular solution Y(t), then the general solution of 
         P0 y(n) + P1 y(n-1) + P2 y(n-2)  + ... + Pn y = G(t)
is 
         y(t) = C1 y1  + C2 y2  + ... + Cn yn+ Y(t)
Homework 4.3: #12, #17
27. Non-homogeneous Equation: Variation of Parameters
Section 4.4		 Suppose the equation:
          P0 y(n) + P1 y(n-1) + P2 y(n-2)  + ... + Pn y = 0
has the general solution:
          y(t) = C1 y1  + C2 y2  + ... + Cn yn

Look for a solution of the equation:
         P0 y(n) + P1 y(n-1) + P2 y(n-2)  + ... + Pn y = G(t)
has the form
         y(t) = u1 y1  + u2 y2  + ... + un yn 
where u1(t), u2(t), ..., un are functions to be determined. 
Homework 4.4: #1, #10

Chapter 5: Series Solutions of Second Order Linear Equations

28. Review of Power Serials
Section 5.1
Homework 5.1: #10, #23
29. Series Solutions: Ordinary Points
Section 5.2 & 5.3		 Consider

            P(x)y'' + Q(x) y' + R(x) y = 0

Definitions: 

  • Ordinary points x0: P(x0) is not zero
  • Singular points x0: P(x0) is zero 

Recurrence Relation

Homework 5.2: #2, #25 (Find the power series solution of 10 terms).
30. Regular Singular Points
Section 5.4

 

 

 

 

Euler Equations

 Consider

            P(x)y'' + Q(x) y' + R(x) y = 0

Definitions: a singular point x0 satisfying
          
            

Consider the equation
            x2y'' + a x y' + b y = 0

Homework 5.4: #11
5.5: #16

Chapter 6: Laplace Transform

31. Laplace Transform and Applications
6.1 & 6.2		 Definition:

Theorem 6.2.1 & Corollary 6.2.2

In Mathematica,
     <<Calculus`LaplaceTransform`

     LaplaceTransform[Sin[t], t, s]

     InverseLaplaceTransform[1/(1+s^2), s, t]

Example 1 & 2 in 6.2

Homework 6.2: #17, #21
32. Systems of First Order Linear Equations
7.1-7.3
Homework