Domain

Details

Differential

• Derivatives or limits to infinity of variables within parameters
• Differentials or small finite changes of variables
• Representing rate of change of parameters w.r.t. to their variables
• Link between phenomenon and model characteristics

Differential equations

• Representing model of the reality in terms of parameters & their derivatives & differentials
• Major classifications are:

1. ODE: ordinary, derivatives of parameters in 1 variable only
2. PDE: partial, derivatives in 2 or more variables
3. Order: the highest derivative inside the DE, indept. of power
4. Homogeneous: all terms are derivatives of parameters, =0 & vice versa
5. Linear: each term is the parameter or its derivatives only
6. Non-linear: combinations of variables, parameters & derivatives
7. Solution types: explicit y=f(x) or implicit F(x,y)
8. Solution forms: no solution, single unique solution (with IC & BC), family of solutions or combinations

1^st order DE

• Exact DE: F(x,y,y')=0
• Solved by integrating factor:

1. Linear: e'''
2. Non-linear: u=y^1-n

• Variable separable: (take note of suppressed solutions =0)

1. Linear: y'= f(x)/g(y)
2. Non-linear: by substitution

• Solve by y=e^mx

2^nd order LDE

• Linearity: POS under addition & scalar multiplication
• Linear independence: using Wrouskian; scalars=0
• Total solution: y= scalar sum of y_h of l.i. components + scalar sum of y_p
• Homogeneous:

1. y=e^mx
2. y=e^px+-iqx
3. Reduction of order: y2=u.y1 with variable coefficients

• Non-homogeneous: f(x)

1. POS of scalar sum of y_p
2. Method of Undetermined Coefficients: y_p
3. Lagrange's variation of parameters:

• let y=sum of v.y_h
• sum of v'.y=0
• sum of v'.y'=f(x)

Higher order LDE

• Use of fundamental solution techniques of 1^st & 2^nd order LDE
• POS of solutions: x = x_h + sum(x_pi)
• n-order DE produces n-parameter family of solutions: G(x,y,C1, …,Cn)=0

Nonlinear ODE

• Non-linear forms:

1. Mixtures of derivatives & y in any one term
2. Fractions of derivatives & y in any one term

• Trial solution forms:

1. Implicit solution, G(x,y)
2. Variable separable
3. Reduction of order: let u=y^1-n & subst. => changing the dependent variables
4. M(x,y) & N(x,y) same degree: let y=u.x or x=v.y
5. Log matching: Guoke(490KB), formulate into log/ln asymptotic equations, evaluate parameter b & express the log matching function

PDE

• Forms: derivatives & terms of multiple dependent variables in multiple independent variables
• Trials solutions: (note convergence, stability, boundedness)

1. Laplace transforms: shifts to algebraic equations for solution before shifting back
2. Fourier transforms
3. Z transforms