1. Chapter 2: Second-Order Differential Equations
2.1. Preliminary Concepts
○ Second-order differential equation
e.g.,
Solution: A function satisfies ,
(I : an interval)

2. ○ Linear second-order differential equation
Nonlinear: e.g.,
2.2. Theory of Solution
○ Consider
y contains two parameters c and d

3. The graph of Given the initial condition

4. Given another initial condition
The graph of
◎ The initial value problem:
○ Theorem 2.1: : continuous on I,
has a unique solution

5. ○ Theorem 2.2: : solutions of Eq. (2.2)
2.2.1.Homogeous Equation
○ Theorem 2.2: : solutions of Eq. (2.2)
solution of Eq. (2.2)
: real numbers
Proof:

6. ※ Two solutions are linearly independent.
Their linear combination provides an
infinity of new solutions
○ Definition 2.1: f , g : linearly dependent
If s.t or ;
otherwise f , g : linearly independent
In other words, f and g are linearly
dependent only if for

7. ○ Wronskian test -- Test whether two solutions of a homogeneous differential equation are linearly independent Define: Wronskian of solutions to be the 2 by 2 determinant

8. ○ Let If : linear dep., then or Assume

9. ○ Theorem 2.3: 1) Either or 2) : linearly independent iff Proof (2): (i) (if : linear indep. (P), then (Q) if ( Q) , then : linear dep. ( P) ) : linear dep.

10. (ii) (if (P), then : linear indep. (Q)
if : linear dep. ( Q), then
( P))
: linear dep.,
※ Test at just one point of I to determine
linear dependency of the solutions

11. 。 Example 2.2:
are solutions of
: linearly independent

12. 。 Example 2.3: Solve by a power series method The Wronskian of at nonzero x would be difficult to evaluate, but at x = 0 are linearly independent

13. ○ Definition 2.2: ◎ Find all solutions 1. : linearly independent
: fundamental set of solutions
: general solution
: constant
○ Theorem 2.4:
: linearly independent solutions on I
Any solution is a linear combination of

14. Proof: Let be a solution.
Show s.t.
Let and
Then, is the unique solution on I of the initial
value problem

15. 2. 2. 2. Nonhomogeneous Equation ○ Theorem 2
Nonhomogeneous Equation ○ Theorem 2.5: : linearly independent homogeneous solutions of : a nonhomogeneous solution of any solution has the form

16. Proof: Given , solutions
: a homogenous solution of
: linearly independent homogenous solutions
(Theorem 2.4)

17. 1. Find the general homogeneous solutions
○ Steps:
1. Find the general homogeneous solutions of
2. Find any nonhomogeneous solution of
3. The general solution of is
2.3. Reduction of Order
-- A method for finding the second independent homogeneous solution when given the first one

18. ○ Let Substituting into ( : a homogeneous solution ) Let (separable)

19. For symlicity, let c = 1, 。 Example 2.4: : a solution Let
: independent solutions
。 Example 2.4:
: a solution
Let

20. Substituting into (A), For simplicity, take c = 1, d = 0 : independent The general solution:

21. 2. 4. Constant Coefficient Homogeneous A, B : numbers ----- (2
2.4. Constant Coefficient Homogeneous A, B : numbers (2.4) The derivative of is a constant (i.e., ) multiple of Constant multiples of derivatives of y , which has form , must sum to 0 for (2,4) ○ Let Substituting into (2,4), (characteristic equation)

22. i) Solutions : : linearly independent The general solution:

23. 。 Example 2.6: Let , Then Substituting into (A), The characteristic equation: The general solution:

24. ii) By the reduction of order method, Let Substituting into (2.4)

25. Choose : linearly independent The general sol. : 。 Example 2
Choose : linearly independent The general sol.: 。 Example 2.7: Characteristic eq. : The repeated root: The general solution:

26. iii) Let The general sol.:

27. 。 Example 2.8: Characteristic equation: Roots: The general solution: ○ Find the real-valued general solution 。 Euler’s formula:

28. Maclaurin expansions:

29. 。 Eq. (2.5),

30. Find any two independent solutions Take
The general sol.:

31. 2.5. Euler’s Equation , A , B : constants -----(2.7)
Transform (2.7) to a constant coefficient equation
by letting

32. Substituting into Eq. (2. 7), i. e. , --------(2
Substituting into Eq. (2.7), i.e., (2.8) Steps: (1) Solve (2) Substitute (3) Obtain

33. Characteristic equation: Roots: General solution:
。 Example 2.11: (A)
(B)
(i) Let
Substituting into (A)
Characteristic equation:
Roots:
General solution:

34. ○ Solutions of constant coefficient linear equation have the forms:
Solutions of Euler’s equation have the forms:

35. 2.6. Nonhomogeneous Linear Equation ------(2.9)
The general solution:
◎ Two methods for finding
(1) Variation of parameters
-- Replace with in the general homogeneous solution
Let
Assume (2.10)
Compute

36. Substituting into (2.9), -----------(2.11) Solve (2.10) and (2.11) for
Likewise,

37. 。 Example 2. 15: ------(A) i) General homogeneous solution : Let
。 Example 2.15: (A) i) General homogeneous solution : Let . Substitute into (A) The characteristic equation: Complex solutions: Real solutions: :independent

38. ii) Nonhomogeneous solution Let

39. iii) The general solution:

40. (2) Undetermined coefficients Apply to A, B: constants Guess the form of from that of R e.g. : a polynomial Try a polynomial for : an exponential for Try an exponential for

41. 。 Example 2.19: ---(A) It’s derivatives can be multiples of or Try Compute Substituting into (A),

42. : linearly independent
and
The homogeneous solutions:
The general solution:

43. 。 Example 2. 20: ------(A) , try Substituting into (A),
。 Example 2.20: (A) , try Substituting into (A), * This is because the guessed contains a homogeneous solution Strategy: If a homogeneous solution appears in any term of , multiply this term by x. If the modified term still occurs in a homogeneous solution, multiply by x again

44. Try Substituting into (A),

45. ○ Steps of undetermined coefficients: (1) Find homogeneous solutions (2) From R(x), guess the form of If a homogeneous solution appears in any term of , multiply this term by x. If the modified term still occurs in a homogeneous solution, multiply by x again (3) Substitute the resultant into and solve for its coefficients

46. ○ Guess from Let : a given polynomial , : polynomials with unknown coefficients
Guessed

47. 2.6.3. Superposition Let be a solution of is a solution of (A)

48. 。 Example 2.25: The general solution: where homogeneous solutions