# 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 Physical laws; Gathering known information; Setting up mathematical equations; Solving the mathematical equations and identifying all parameters; 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 ynLook 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,      <