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First app. of laplace equation
To solve first order equation.
dx(t)/dt=Ax(t)+b(t)→x(t)=L⁻¹{(sI-A)⁻¹(v+c(s)}.
Second application of laplace equation
exp(tA)
L⁻¹([sI-A]⁻¹).
When f(t)=δ(t),w(t)=y(t).
Weighting function for a₀dⁿy/dxⁿ+...+aⁿy=f(t)
y(t)=0∫tw(u)f(t-u)du.
y(t) is equal to by weighting function method
L{t^mfⁿ(t)}
(-1)^mDm[sⁿF(s)-...-fⁿ⁻¹(0)].
EId⁴y/dx⁴=w(x).
What is the beam equation supporting a point load?
M/a+Qδ(x-c).
Distribution of weight across beam clamped at one point
xy''+(1-x)y'+ny=0.
Laguerre equation
y''-2xy'+2ny=0.
Hermite equation
L(tf(t))
-F'(s).
L(tf'(t))
-sF'(s)-F(s).
L(tf''(t))
-s²F'(s)-2sF(s)+f(0).
Control systems
Systems that compare between an input and output attempt to make it zero.
U₀(s)/Ui(s).
The transfer function
Stability.
When a bounded input results in bounded output
Test of stability.
Denominator of G(s) has roots negative real part
Steering mechanisms.
Input angle θi→amplifier for error signal→resisting torque producing amplifier torque K(θi-θo) and resistive torque -kdθo/dt.
(1-x²)y''-2xy'+α(α+1)y=0.
Legendre equation.
Pn(1)=1.
Legendre polynomial scaling
P2n(x)
0ton∑(-1)^r(4n-2r)!x^(2n-2r)/2²ⁿr!(2n-r)!(2n-2r)!.
P2n+1(x)
0ton∑(-1)^r(4n-2r+2)!x^(2n-2r+1)/2²ⁿ⁺¹r!(2n-r+1)!(2n-2r+1)!.
(n+1)Pn+1(x)-(2n+1)Pn(x)+nPn-1(x)=0.
Recurrence relation
Ordinary,singular,non-singular.
Types of mind
y(x)=(x-x₀)^c0to∞∑an(x-x₀)ⁿ.
Frobenius method
f(x)=a₀+1to∞∑(ancosnx+bnsinx).
Fourier series
a₀
-π∫πf(x)dx/2π.
an
-π∫πf(x)cosnxdx/π.
AU=aU-bV,AV=bU+aV
A be a real matrix with non real eigenvalues λ=a±ib and X=U±iV
[a,b;-b,a]
In 29 P=[U|V] P⁻¹AP=
P₀+tX
Line in matrix form
P₀+sX+tY
Plane in matrix form
AP.(ABxAC)=0
With three points A,B,C non-collinear
X^tY
X.Y
|X.Y|≤||X||.||Y||
Cauchy-Schwarz inequality
B A P
In Joachimsthal's ratio formulae |t|=AP/AB if t>1_is between_and_.
A P B
In Joachimsthal's ratio formulae |t|=AP/AB if t<0_is between_and_.
F(s)=0⁻∫∞f(t)exp(-st)dt
Laplace transform
F+G
L(f+g)
s^kF(s)-s^k-1f(0)s^k-2df(0)/dt...
d^kf/dt^k=_______.
G(s)=F(s)/s
g(t)=0∫tf(tau)dtau
F(s/α)/α
f(αt)
F(s-a)
exp(at)f(t)
(-1)^kd^kF(s)/ds^k
t^kf(t)
s∫∞F(s)ds
f(t)/t
1/s^k+1
t^k
s/s²+w²
coswt
w/s²+w²
sinwt
exp(-as)
δ(t-a)
s/s²-k²
cosh(kt)
k/s²-k²
sinh(kt)
1/√s
1/√(πt)
Γ(a+1)/s^a+1
t^a
1/(s²+w²)²
(sin(wt)-wtcos(wt))/2w³
s/(s²+w²)²
tsin(wt)/2w
s²/(s²+w²)²
(sin(wt)+wtcos(wt))/2w
exp(-as)/s
u(t-a)