[工數] 勝仕班考試重點提醒 by SSW
Check list for Exam 1 of Engineering Mathematics:
Chapter 1
1. ODE and PDE
2. Order, degree of ODE
3. Linear and nonlinear
4. Homogeneous and non-homogeneous
5. Types of solutions of ODEs: general solution (G.S.), particular solution
(P.S.), singular solution (S.S.)
6. Singular solution and envelop
7. Initial Value Problem (IVP) and boundary value problem (BVP)
8. Separable type of ODE, separation of variables
9. Types of ODEs that can be converted into separable type of ODEs (two lines
with intersection, or two parallel lines)
10.Exact ODEs: definition of exact, theorem of exact, how to solve exact ODEs
11.Non-exact: how to find integrating factor (I.F.), the meaning of I.F.
12.Standard form of first order, linear ODEs: how to solve? (3 methods)
13.How to solve Bernoulli’s equations and Ricatti equations
14.Existence and uniqueness of solutions for first order ODEs
15.Modeling the mixing tanks (conservation of mass)
Chapter 2
1. Differential operator (D = d/dx) and its properties
2. Definition of linear combination of functions
3. Definitions of linearly dependent (L.D.) and linearly independent (L.I.)
set of functions
4. Definition of Wronskian determinant (you should also know Cramer’s rule),
when and why do we need it?
5. Important theorem of the solution for homogeneous ODEs (what is the
general solution?, superposition principle)
6. Important theorem of the solution for non-homogeneous ODEs (what is the
general solution?)
7. The general procedure of solving any order of non-homogeneous linear ODEs.
8. How to solve the second or higher order, linear, homogeneous ODEs with
constant coefficients? (two distinct roots, repeated roots, conjugated
roots): (A) Aspect I: try y = exp(λx) (B) Aspect II: use differential
operator (D) and the concept from Chapter 1.
9. How many methods can we use to find the particular solution for any order
of non-homogeneous linear ODEs? (Undeterminded coefficients, variation of
parameters, inverse operator)
Good luck!!
SORRY 我排版有點亂:P
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10/26 23:31, , 1F
10/26 23:31, 1F