ingenieria técnica en llum y so y calor

1r curs de electrónica analógica

i^{2} = (-1)

q(t) = pe^{at} || q(t) = pe^{(-1)·at}

q(t) = pe^{i·at} || q(t) = pe^{(-i)·at}

2n curs de electrónica virtual

k^{2} = i

j^{2} = (-i)

q(t) = pe^{k·at} || q(t) = pe^{(-k)·at}

q(t) = pe^{j·at} || q(t) = pe^{(-j)·at}

Teoría matemàtica:

Teorema:

d_{t}[ f(t)+g(t) ] = d_{t}[f(t)]+d_{t}[g(t)]

d_{t}[ s·f(t) ] = s·d_{t}[f(t)]

Teorema:

d_{t}[g( f(t) )] = d_{f(t)}[g( f(t) )]·d_{t}[f(t)]

Teorema:

d_{t}[t] = 1

d_{t}[e^{t}] = e^{t}

Teorema:

d_{t}[e^{at}] = ae^{at}

d_{tt}^{2}[e^{at}] = a^{2}·e^{at}

Teorema:

d_{tt}^{2}[ f(t)+g(t) ] = d_{tt}^{2}[f(t)]+d_{tt}^{2}[g(t)]

Teorema:

int[s]d[t] = st

Teorema:

d_{t}[ ln(f(t)) ] = ( d_{t}[f(t)]/f(t) )

Teorema:

d_{t}[ f(t) [o(t)o] g(t) ] = d_{t}[f(t)]·d_{t}[g(t)]

Teoría física:

1r curs de electrónica analógica:

Osciladors:

Lley:

R·d_{t}[q(t)]+(-C)·q(t) = 0

q(t) = pe^{(C/R)·t}

R·d_{t}[q(t)]+C·q(t) = 0

q(t) = pe^{(-1)·(C/R)·t}

Lley:

R·d_{t}[q(t)]+i·(-C)·q(t) = 0

q(t) = pe^{i·(C/R)·t}

R·d_{t}[q(t)]+i·C·q(t) = 0

q(t) = pe^{(-i)·(C/R)·t}

Bifurcadors:

Lley:

L·d_{tt}^{2}[q(t)]+(-C)·q(t) = 0

q(t) = pe^{(C/L)^{(1/2)}·t} || q(t) = pe^{(-1)·(C/L)^{(1/2)}·t}

L·d_{tt}^{2}[q(t)]+C·q(t) = 0

q(t) = pe^{i·(C/L)^{(1/2)}·t} || q(t) = pe^{(-i)·(C/L)^{(1/2)}·t}

Lley:

L·d_{tt}^{2}[f(t)+g(t)]+(-C)·(f(t)+g(t)) = 0

f(t) = pe^{(C/L)^{(1/2)}·t} & g(t) = pe^{(-1)·(C/L)^{(1/2)}·t}

L·d_{tt}^{2}[f(t)+g(t)]+C·(f(t)+g(t)) = 0

f(t) = pe^{i·(C/L)^{(1/2)}·t} & g(t) = pe^{(-i)·(C/L)^{(1/2)}·t}

Distorsoniados:

Lley:

R·d_{t}[q(t)]+(-C)·q(t) = Ae^{st}

q(t) = A·( 1/(s·R+(-C)) )·e^{st}

R·d_{t}[q(t)]+C·q(t) = Ae^{(-1)·st}

q(t) = A·( 1/((-s)·R+C) )·e^{(-1)·st}

Lley:

R·d_{t}[q(t)]+i·(-C)·q(t) = Ae^{i·st}

q(t) = A·(1/i)·( 1/(s·R+(-C)) )·e^{i·st}

R·d_{t}[q(t)]+i·C·q(t) = Ae^{(-i)·st}

q(t) = A·(1/i)·( 1/((-s)·R+C) )·e^{(-i)·st}

Amplificadors:

Lley:

L·d_{tt}^{2}[q(t)]+(-C)·q(t) = Ae^{st}

q(t) = A·( 1/(s^{2}·L+(-C)) )·e^{st}

L·d_{tt}^{2}[q(t)]+(-C)·q(t) = Ae^{(-1)·st}

q(t) = A·( 1/(s^{2}·L+(-C)) )·e^{(-1)·st}

Lley:

L·d_{tt}^{2}[q(t)]+C·q(t) = Ae^{i·st}

q(t) = A·( 1/((-1)·s^{2}·L+C) )·e^{i·st}

L·d_{tt}^{2}[q(t)]+C·q(t) = Ae^{(-i)·st}

q(t) = A·( 1/((-1)·s^{2}·L+C) )·e^{(-i)·st}

Lley:

n resistencies en serie <==> R = ( R_{1}+...+R_{n} )

n resistencies en paralel <==> R = ( (1/R_{1})+...+(1/R_{n}) )^{(-1)}

Lley:

R = ( R_{1}+...+R_{n} )·( (1/R_{1})+...+(1/R_{m}) )·( R_{1}+...+R_{n} )

R = ( ( (1/R_{1})+...+(1/R_{n}) )·( R_{1}+...+R_{m} )·( (1/R_{1})+...+(1/R_{n}) ) )^{(-1)}

Principi:

E(x) = qk·(1/r^{2})·(x/r)

B(x) = (-1)·qk·(1/r^{2})·(d_{t}[x]/r)

Lley:

m·d_{tt}^{2}[h(t)] = p·( E( h(t) )+int[ B( h(t) ) ]d[t] ) = 0

h(t) = ct

(-1)·h(t) = (-c)·t

Microfons y Altavoxums mono:

Lley:

int[ c·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{(C/R)·t}

int[ (-c)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{(C/R)·t}

Lley:

int[ (-c)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{(-1)·(C/R)·t}

int[ c·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{(-1)·(C/R)·t}

Cámares y Pantalles mono:

Lley:

int[ c·(1/i)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{i·(C/R)·t}

int[ (-c)·(1/i)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{i·(C/R)·t}

Lley:

int[ (-c)·(1/i)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{(-i)·(C/R)·t}

int[ c·(1/i)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{(-i)·(C/R)·t}

Microfons y Altavoxums stereos:

Lley:

int[ c·(L/C)·( d_{tt}^{2}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{(C/L)^{(1/2)}·t} || q(t) = pe^{(-1)·(C/L)^{(1/2)}·t}

int[ (-c)·(L/C)·( d_{tt}^{2}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{(C/L)^{(1/2)}·t} || q(t) = pe^{(-1)·(C/L)^{(1/2)}·t}

Lley:

int[ c·(L/C)·( d_{tt}^{2}[f(t)+g(t)]/(f(t)+g(t)) ) ]d[t] = h(t)

f(t) = pe^{(C/L)^{(1/2)}·t} & g(t) = pe^{(-1)·(C/L)^{(1/2)}·t}

int[ (-c)·(L/C)·( d_{tt}^{2}[f(t)+g(t)]/(f(t)+g(t)) ) ]d[t] = (-1)·h(t)

f(t) = pe^{(C/L)^{(1/2)}·t} & g(t) = pe^{(-1)·(C/L)^{(1/2)}·t}

Cámares y Pantalles stereos:

Lley:

int[ (-c)·(L/C)·( d_{tt}^{2}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{i·(C/L)^{(1/2)}·t} || q(t) = pe^{(-i)·(C/L)^{(1/2)}·t}

int[ c·(L/C)·( d_{tt}^{2}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{i·(C/L)^{(1/2)}·t} || q(t) = pe^{(-i)·(C/L)^{(1/2)}·t}

Lley:

int[ (-c)·(L/C)·( d_{tt}^{2}[f(t)+g(t)]/(f(t)+g(t)) ) ]d[t] = h(t)

f(t) = pe^{i·(C/L)^{(1/2)}·t} & g(t) = pe^{(-i)·(C/L)^{(1/2)}·t}

int[ c·(L/C)·( d_{tt}^{2}[f(t)+g(t)]/(f(t)+g(t)) ) ]d[t] = (-1)·h(t)

f(t) = pe^{i·(C/L)^{(1/2)}·t} & g(t) = pe^{(-i)·(C/L)^{(1/2)}·t}

Reproductors de disc:

Lley:

h(t) = ( ln(1/p)+ln(q(t)) ) [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (C/R)·r(t)

q(t) = pe^{(C/R)·t}

h(t) = ( ln(1/p)+ln(q(t)) ) [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (-1)·(C/R)·r(t)

q(t) = pe^{(-1)·(C/R)·t}

Lley:

h(t) = ( ln(1/p)+ln(q(t)) ) [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = i·(C/R)·r(t)

q(t) = pe^{i·(C/R)·t}

h(t) = ( ln(1/p)+ln(q(t)) ) [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (-i)·(C/R)·r(t)

q(t) = pe^{(-i)·(C/R)·t}

Reproductor de disc amb pitch:

Lley:

h(t) = int[ (C/L)^{(1/2)}·( ln(1/p)+ln(q(t)) ) ]d[t] [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (C/L)·t·r(t)

d_{tt}^{2}[h(t)] = (C/L)·r(t)+(C/L)·t·d_{t}[r(t)]

q(t) = pe^{(C/L)^{(1/2)}·t}

h(t) = int[ (C/L)^{(1/2)}·( ln(1/p)+ln(q(t)) ) ]d[t] [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (-1)·(C/L)·t·r(t)

d_{tt}^{2}[h(t)] = (-1)·(C/L)·r(t)+(-1)·(C/L)·t·d_{t}[r(t)]

q(t) = pe^{(-1)·(C/L)^{(1/2)}·t}

Lley:

h(t) = int[ (C/L)^{(1/2)}·( ln(1/p)+ln(q(t)) ) ]d[t] [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = i·(C/L)·t·r(t)

d_{tt}^{2}[h(t)] = i·(C/L)·r(t)+i·(C/L)·t·d_{t}[r(t)]

q(t) = pe^{i·(C/L)^{(1/2)}·t}

h(t) = int[ (C/L)^{(1/2)}·( ln(1/p)+ln(q(t)) ) ]d[t] [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (-i)·(C/L)·t·r(t)

d_{tt}^{2}[h(t)] = (-i)·(C/L)·r(t)+(-i)·(C/L)·t·d_{t}[r(t)]

q(t) = pe^{(-i)·(C/L)^{(1/2)}·t}

Pitch:

Lley:

P = < R_{1},L_{1},...,R_{n},L_{n}>

Q = < L_{1},R_{1},...,L_{n},R_{n}>

2n curs de electrónica virtual:

Osciladors virtuals:

Lley:

R·d_{t}[q(t)]+k·(-C)·q(t) = 0

q(t) = pe^{k·(C/R)·t}

R·d_{t}[q(t)]+k·C·q(t) = 0

q(t) = pe^{(-k)·(C/R)·t}

Lley:

R·d_{t}[q(t)]+j·(-C)·q(t) = 0

q(t) = pe^{j·(C/R)·t}

R·d_{t}[q(t)]+j·C·q(t) = 0

q(t) = pe^{(-j)·(C/R)·t}

Bifurcadors virtuals:

Lley:

L·d_{tt}^{2}[q(t)]+i·(-C)·q(t) = 0

q(t) = pe^{k·(C/L)^{(1/2)}·t} || q(t) = pe^{(-k)·(C/L)^{(1/2)}·t}

L·d_{tt}^{2}[q(t)]+i·C·q(t) = 0

q(t) = pe^{j·(C/L)^{(1/2)}·t} || q(t) = pe^{(-j)·(C/L)^{(1/2)}·t}

Lley:

L·d_{tt}^{2}[f(t)+g(t)]+i·(-C)·(f(t)+g(t)) = 0

f(t) = pe^{k·(C/L)^{(1/2)}·t} & g(t) = pe^{(-k)·(C/L)^{(1/2)}·t}

L·d_{tt}^{2}[f(t)+g(t)]+i·C·(f(t)+g(t)) = 0

f(t) = pe^{j·(C/L)^{(1/2)}·t} & g(t) = pe^{(-j)·(C/L)^{(1/2)}·t}

Distorsoniados virtuals:

Lley:

R·d_{t}[q(t)]+k·(-C)·q(t) = Ae^{k·st}

q(t) = A·(1/k)·( 1/(s·R+(-C)) )·e^{k·st}

R·d_{t}[q(t)]+k·C·q(t) = Ae^{(-k)·st}

q(t) = A·(1/k)·( 1/((-s)·R+C) )·e^{(-k)·st}

Lley:

R·d_{t}[q(t)]+j·(-C)·q(t) = Ae^{j·st}

q(t) = A·(1/j)·( 1/(s·R+(-C)) )·e^{j·st}

R·d_{t}[q(t)]+j·C·q(t) = Ae^{(-j)·st}

q(t) = A·(1/j)·( 1/((-s)·R+C) )·e^{(-j)·st}

Amplificadors virtuals:

Lley:

L·d_{tt}^{2}[q(t)]+i·(-C)·q(t) = Ae^{k·st}

q(t) = A·(1/i)·( 1/(s^{2}·L+(-C)) )·e^{k·st}

L·d_{tt}^{2}[q(t)]+i·(-C)·q(t) = Ae^{(-k)·st}

q(t) = A·(1/i)·( 1/(s^{2}·L+(-C)) )·e^{(-k)·st}

Lley:

L·d_{tt}^{2}[q(t)]+i·C·q(t) = Ae^{j·st}

q(t) = A·(1/i)·( 1/((-1)·s^{2}·L+C) )·e^{j·st}

L·d_{tt}^{2}[q(t)]+i·C·q(t) = Ae^{(-j)·st}

q(t) = A·(1/i)·( 1/((-1)·s^{2}·L+C) )·e^{(-j)·st}

Microfons y Altavoxums virtuals mono:

Lley:

int[ c·(1/k)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{k·(C/R)·t}

int[ (-c)·(1/k)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{k·(C/R)·t}

Lley:

int[ (-c)·(1/k)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{(-k)·(C/R)·t}

int[ c·(1/k)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{(-k)·(C/R)·t}

Cámares y Pantalles virtuals mono:

Lley:

int[ c·(1/j)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{j·(C/R)·t}

int[ (-c)·(1/j)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{j·(C/R)·t}

Lley:

int[ (-c)·(1/j)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{(-j)·(C/R)·t}

int[ c·(1/j)·(R/C)·( d_{t}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{(-j)·(C/R)·t}

Microfons y Altavoxums virtuals stereos:

Lley:

int[ c·(1/i)·(L/C)·( d_{tt}^{2}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{k·(C/L)^{(1/2)}·t} || q(t) = pe^{(-k)·(C/L)^{(1/2)}·t}

int[ (-c)·(1/i)·(L/C)·( d_{tt}^{2}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{k·(C/L)^{(1/2)}·t} || q(t) = pe^{(-k)·(C/L)^{(1/2)}·t}

Lley:

int[ c·(1/i)·(L/C)·( d_{tt}^{2}[f(t)+g(t)]/(f(t)+g(t)) ) ]d[t] = h(t)

f(t) = pe^{k·(C/L)^{(1/2)}·t} & g(t) = pe^{(-k)·(C/L)^{(1/2)}·t}

int[ (-c)·(1/i)·(L/C)·( d_{tt}^{2}[f(t)+g(t)]/(f(t)+g(t)) ) ]d[t] = (-1)·h(t)

f(t) = pe^{k·(C/L)^{(1/2)}·t} & g(t) = pe^{(-k)·(C/L)^{(1/2)}·t}

Cámares y Pantalles virtuals stereos:

Lley:

int[ (-c)·(1/i)·(L/C)·( d_{tt}^{2}[q(t)]/q(t) ) ]d[t] = h(t)

q(t) = pe^{j·(C/L)^{(1/2)}·t} || q(t) = pe^{(-j)·(C/L)^{(1/2)}·t}

int[ c·(1/i)·(L/C)·( d_{tt}^{2}[q(t)]/q(t) ) ]d[t] = (-1)·h(t)

q(t) = pe^{j·(C/L)^{(1/2)}·t} || q(t) = pe^{(-j)·(C/L)^{(1/2)}·t}

Lley:

int[ (-c)·(1/i)·(L/C)·( d_{tt}^{2}[f(t)+g(t)]/(f(t)+g(t)) ) ]d[t] = h(t)

f(t) = pe^{j·(C/L)^{(1/2)}·t} & g(t) = pe^{(-j)·(C/L)^{(1/2)}·t}

int[ c·(1/i)·(L/C)·( d_{tt}^{2}[f(t)+g(t)]/(f(t)+g(t)) ) ]d[t] = (-1)·h(t)

f(t) = pe^{j·(C/L)^{(1/2)}·t} & g(t) = pe^{(-j)·(C/L)^{(1/2)}·t}

Reproductors de disc virtuals:

Lley:

h(t) = ( ln(1/p)+ln(q(t)) ) [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = k·(C/R)·r(t)

q(t) = pe^{k·(C/R)·t}

h(t) = ( ln(1/p)+ln(q(t)) ) [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (-k)·(C/R)·r(t)

q(t) = pe^{(-k)·(C/R)·t}

Lley:

h(t) = ( ln(1/p)+ln(q(t)) ) [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = j·(C/R)·r(t)

q(t) = pe^{j·(C/R)·t}

h(t) = ( ln(1/p)+ln(q(t)) ) [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (-j)·(C/R)·r(t)

q(t) = pe^{(-j)·(C/R)·t}

Reproductor de disc virtuals amb pitch:

Lley:

h(t) = int[ (C/L)^{(1/2)}·( ln(1/p)+ln(q(t)) ) ]d[t] [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = k·(C/L)·t·r(t)

d_{tt}^{2}[h(t)] = k·(C/L)·r(t)+k·(C/L)·t·d_{t}[r(t)]

q(t) = pe^{k·(C/L)^{(1/2)}·t}

h(t) = int[ (C/L)^{(1/2)}·( ln(1/p)+ln(q(t)) ) ]d[t] [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (-k)·(C/L)·t·r(t)

d_{tt}^{2}[h(t)] = (-k)·(C/L)·r(t)+(-k)·(C/L)·t·d_{t}[r(t)]

q(t) = pe^{(-k)·(C/L)^{(1/2)}·t}

Lley:

h(t) = int[ (C/L)^{(1/2)}·( ln(1/p)+ln(q(t)) ) ]d[t] [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = j·(C/L)·t·r(t)

d_{tt}^{2}[h(t)] = j·(C/L)·r(t)+j·(C/L)·t·d_{t}[r(t)]

q(t) = pe^{j·(C/L)^{(1/2)}·t}

h(t) = int[ (C/L)^{(1/2)}·( ln(1/p)+ln(q(t)) ) ]d[t] [o(t)o] int[ r(t) ]d[t]

d_{t}[h(t)] = (-j)·(C/L)·t·r(t)

d_{tt}^{2}[h(t)] = (-j)·(C/L)·r(t)+(-j)·(C/L)·t·d_{t}[r(t)]

q(t) = pe^{(-j)·(C/L)^{(1/2)}·t}

3r curs de tecnología del calor:

T(x) = (R·q)(x)

T(x) = (P+(-Q))(x)

Increment de temperatura <==> T(x) >] 0

Decrement de temperatura <==> T(x) [< 0

Lley:

v·d_{x}[T(x)]+(-u)·T(x) = 0

T(x) = we^{(u/v)·x}

v·d_{x}[T(x)]+u·T(x) = 0

T(x) = we^{(-1)·(u/v)·x}

Lley:

v·d_{x}[T(x)]+i·(-u)·T(x) = 0

T(x) = we^{i·(u/v)·x}

v·d_{x}[T(x)]+i·u·T(x) = 0

T(x) = we^{(-i)·(u/v)·x}

Lley:

(h/m)·d_{xx}^{2}[T(x)]+(-u)·T(x) = 0

T(x) = we^{( u·(m/h) )^{(1/2)}·x} || T(x) = we^{(-1)·( u·(m/h) )^{(1/2)}·x}

(h/m)·d_{xx}^{2}[T(x)]+u·T(x) = 0

T(x) = we^{i·( u·(m/h) )^{(1/2)}·x} || T(x) = we^{(-i)·( u·(m/h) )^{(1/2)}·x}

Lley:

(h/m)·d_{xx}^{2}[f(x)+g(x)]+(-u)·(f(x)+g(x)) = 0

f(x) = we^{( u·(m/h) )^{(1/2)}·x} & g(x) = we^{(-1)·( u·(m/h) )^{(1/2)}·x}

(h/m)·d_{xx}^{2}[f(x)+g(x)]+u·(f(x)+g(x)) = 0

f(x) = we^{i·( u·(m/h) )^{(1/2)}·x} & g(x) = we^{(-i)·( u·(m/h) )^{(1/2)}·x}

Lley:

v·d_{x}[T(x)]+(-u)·T(x) = Ae^{ax}

T(x) = A·( 1/(av+(-u)) )·e^{ax}

v·d_{x}[T(x)]+u·T(x) = Ae^{(-1)·ax}

T(x) = A·( 1/((-a)·v+u) )·e^{(-1)·ax}

Lley:

v·d_{x}[T(x)]+i·(-u)·T(x) = Ae^{i·ax}

T(x) = A·(1/i)·( 1/(av+(-u)) )·e^{i·ax}

v·d_{x}[T(x)]+i·u·T(x) = Ae^{(-i)·ax}

T(x) = A·(1/i)·( 1/((-a)·v+u) )·e^{(-i)·ax}

Lley:

(h/m)·d_{xx}^{2}[T(x)]+(-u)·T(x) = Ae^{ax}

T(x) = A·( 1/(a^{2}·(h/m)+(-u)) )·e^{ax}

(h/m)·d_{xx}^{2}[T(x)]+(-u)·T(x) = Ae^{(-1)·ax}

T(x) = A·( 1/(a^{2}·(h/m)+(-u)) )·e^{(-1)·ax}

Lley:

(h/m)·d_{xx}^{2}[T(x)]+u·T(x) = Ae^{i·ax}

T(x) = A·( 1/((-1)·a^{2}·(h/m)+u) )·e^{i·ax}

(h/m)·d_{xx}^{2}[T(x)]+u·T(x) = Ae^{(-i)·ax}

T(x) = A·( 1/((-1)·a^{2}·(h/m)+u) )·e^{(-i)·ax}

Lley:

int[ c·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = h(t)

T(x) = we^{(u/v)·x}

int[ (-c)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = (-1)·h(t)

T(x) = we^{(u/v)·x}

Lley:

int[ (-c)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = h(t)

T(x) = we^{(-1)·(u/v)·x}

int[ c·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = (-1)·h(t)

T(x) = we^{(-1)·(u/v)·x}

Lley:

int[ c·(1/i)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = h(t)

T(x) = we^{i·(u/v)·x}

int[ (-c)·(1/i)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = (-1)·h(t)

T(x) = we^{i·(u/v)·x}

Lley:

int[ (-c)·(1/i)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = h(t)

T(x) = we^{(-i)·(u/v)·x}

int[ c·(1/i)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = (-1)·h(t)

T(x) = we^{(-i)·(u/v)·x}

4r curs de tecnología virtual del calor:

Lley:

v·d_{x}[T(x)]+k·(-u)·T(x) = 0

T(x) = we^{k·(u/v)·x}

v·d_{x}[T(x)]+k·u·T(x) = 0

T(x) = we^{(-k)·(u/v)·x}

Lley:

v·d_{x}[T(x)]+j·(-u)·T(x) = 0

T(x) = we^{j·(u/v)·x}

v·d_{x}[T(x)]+j·u·T(x) = 0

T(x) = we^{(-j)·(u/v)·x}

Lley:

(h/m)·d_{xx}^{2}[T(x)]+i·(-u)·T(x) = 0

T(x) = we^{k·( u·(m/h) )^{(1/2)}·x} || T(x) = we^{(-k)·( u·(m/h) )^{(1/2)}·x}

(h/m)·d_{xx}^{2}[T(x)]+i·u·T(x) = 0

T(x) = we^{j·( u·(m/h) )^{(1/2)}·x} || T(x) = we^{(-j)·( u·(m/h) )^{(1/2)}·x}

Lley:

(h/m)·d_{xx}^{2}[f(x)+g(x)]+i·(-u)·(f(x)+g(x)) = 0

f(x) = we^{k·( u·(m/h) )^{(1/2)}·x} & g(x) = we^{(-k)·( u·(m/h) )^{(1/2)}·x}

(h/m)·d_{xx}^{2}[f(x)+g(x)]+i·u·(f(x)+g(x)) = 0

f(x) = we^{j·( u·(m/h) )^{(1/2)}·x} & g(x) = we^{(-j)·( u·(m/h) )^{(1/2)}·x}

Lley:

v·d_{x}[T(x)]+k·(-u)·T(x) = Ae^{k·ax}

T(x) = A·(1/k)·( 1/(av+(-u)) )·e^{k·ax}

v·d_{x}[T(x)]+k·u·T(x) = Ae^{(-k)·ax}

T(x) = A·(1/k)·( 1/((-a)·v+u) )·e^{(-k)·ax}

Lley:

v·d_{x}[T(x)]+j·(-u)·T(x) = Ae^{j·ax}

T(x) = A·(1/j)·( 1/(av+(-u)) )·e^{j·ax}

v·d_{x}[T(x)]+j·u·T(x) = Ae^{(-j)·ax}

T(x) = A·(1/j)·( 1/((-a)·v+u) )·e^{(-j)·ax}

Lley:

(h/m)·d_{xx}^{2}[T(x)]+i·(-u)·T(x) = Ae^{k·ax}

T(x) = A·(1/i)·( 1/(a^{2}·(h/m)+(-u)) )·e^{k·ax}

(h/m)·d_{xx}^{2}[T(x)]+i·(-u)·T(x) = Ae^{(-k)·ax}

T(x) = A·(1/i)·( 1/(a^{2}·(h/m)+(-u)) )·e^{(-k)·ax}

Lley:

(h/m)·d_{xx}^{2}[T(x)]+i·u·T(x) = Ae^{j·ax}

T(x) = A·(1/i)·( 1/((-1)·a^{2}·(h/m)+u) )·e^{j·ax}

(h/m)·d_{xx}^{2}[T(x)]+i·u·T(x) = Ae^{(-j)·ax}

T(x) = A·(1/i)·( 1/((-1)·a^{2}·(h/m)+u) )·e^{(-j)·ax}

Lley:

int[ c·(1/k)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = h(t)

T(x) = we^{k·(u/v)·x}

int[ (-c)·(1/k)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = (-1)·h(t)

T(x) = we^{k·(u/v)·x}

Lley:

int[ (-c)·(1/k)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = h(t)

T(x) = we^{(-k)·(u/v)·x}

int[ c·(1/k)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = (-1)·h(t)

T(x) = we^{(-k)·(u/v)·x}

Lley:

int[ c·(1/j)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = h(t)

T(x) = we^{j·(u/v)·x}

int[ (-c)·(1/j)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = (-1)·h(t)

T(x) = we^{j·(u/v)·x}

Lley:

int[ (-c)·(1/j)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = h(t)

T(x) = we^{(-j)·(u/v)·x}

int[ c·(1/j)·(v/u)·( d_{x}[T(x)]/T(x) ) ]d[t] = (-1)·h(t)

T(x) = we^{(-j)·(u/v)·x}