Basics
Engineering
Prerequisites
0.1- exponential, log
- sin, cos, identities
- even, odd functions
- tan
- cosh, sinh
0.2
- First order Partial Derivatives of f(x,y)
- 2nd order partial derivatives
- f(x,y,z) 1st and 2nd partial derivatives
0.3
- System of equations
- Determinants of second order
- elements, row, columns
- cramer's rule
- homogeneous/non-homogeneous, trivial solutions
- determinants of third order
- existence of solutions, no.of solutions
- minor, co-factor
- properties of determinants
0.4
- complex numbers, real/imaginary parts
- cartesian coordinate system
- complex plane/argand diagram
- equality / addition / subtraction / multiply / division
- conjugate
- commutative, assosiative, etc laws
0.5
- polar form of complex numbers/ trignometric form
- absolute value, argument
- usefulness in multiplication, division
- properties in multiplication, division
0.6
- numerical computation
- absolute error
- relative error
- rounding error, guarding figures
0.7
- solutions of equations f(x) = 0
- quadratic equations
- newton's method
- method of false position (regula falsi)
- iteration method
0.8
1.1
- ode vs. partial differential equations
- order of an ODE
- solution of first-order ODE
- implicit, explicit solution
- y = g(x), G(x,y) = 0
- general, particular solution
- singular solutions
- not important for engineering
- solutions which are linear
1.2
- implicit form of ODE
- F(x,y,y') = 0
- explicit form of ODE
- y' = f(x,y)
- isoclines
- curves of constant slope
- f(x,y) = constant
- direction field
- perpendicular to isoclines
- Seperable equations
- g(y)y' = f(x)
- g(y)dy = f(x)dx
- integrate both sides
- initial value problems, initial conditions
1.4
- Reducable to seperable equations
- change of variables
- y' = g(y/x)
- y/x = u, y' = u+u'x
- u + u'x = g(u)
- du/(g(u)-u) = dx/x
- integrate both sides
1.5
- exact differential equations
- definition of exact
- M(x,y)dx + N(x,y)dy = 0
- du = pdu/pdx*du + pdu/pdy*dy
- du = 0
- integrate both sides
1.6
- Integrating factors
- P(x,y)dx + Q(x,y)dy = 0
- make exact by multiplying with F(x)
1.7
- Linear ODE
- Linear form
- y' + f(x)y = r(x)
- homogeneous/non-homogeneous
- homogeneous form solution
- non-homogeneous solution
- input, output/response of system
- (image: block diagram of system with input and output)
1.8
- variation of parameters
- y' + f(x)y = r(x)
- y(x) = u(x)r(x)
- voltage-resistor circuit
- RL, RC circuit
1.10
- one-parameter family of curves
- F(x,y,c) = 0 for fixed value of c
- represents a curve in x-y plane
- parameter of family
- representation using differential equation y' = f(x,y)
- orthogonal trajectories
1.11
- Picard's iteration method
- approximate solution of IVP
- Existence, uniqueness of solutions
- existence theorem (solutions of IVP)
- uniqueness theorem (unique solution)
1.13
useful for solving linear differential equations
4.1
- 3 steps for solving equations with laplace transform
- subsidiary equations
- advantages
- F(s) = L{f} = integral(0->inf)e^(-st)f(t)dt
- f(t) = L-1{f}
- linearity property
- af(t) + bg(t) --> aF(s) + bG(s)
- LT of some elementary functions
- first shifting theorem
- e^at*f(t) --> F(s-a)
- existence theorem
- sufficient conditions for existence of laplace transform of function
- 1. piecewise continuous
- 2. increase as t approaches infinity
- uniqueness of function and its LT
4.2
- diff/integration -> multiplication/division
- differentiation theorem
- f' --> sF(s) - f(0)
- conditions
- derivative of order n
- f'(n) --> s^nF(s) - s^(n-1)f(0) - s^(n-2)f'(o) - ... - f'(n-1)(0)
- existence theorem
- integration of f(t)
- int(0->t) of f(t) dt --> F(s)/s
