ecuació diferencial

x^{2}·d_{xx}^{2}[y(x)]+x·d_{x}[y(x)] = m^{2}·y(x)

y(x) = x^{m}

x^{2}·d_{xx}^{2}[y(x)]+x·d_{x}[y(x)] = ∑ c_{n}·x^{n}

y(x) = ∑ ( c_{n}·( 1/(n^{2}) )·x^{n} )

ecuació diferencial

x·d_{xx}^{2}[y(x)]+d_{x}[y(x)] = y(x)

y(x) = ∑ ( ( 1/(n^{2})! )·x^{n} ) = ∑ ( x^{i(n)} )

p+1 = n <==> ∑ ( x^{i(n+(-1))} ) = ∑ ( x^{i(p)} )

x·d_{xx}^{2}[y(x)]+d_{x}[y(x)] = ∑ ( c_{n}·x^{n} )

y(x) = ∑ ( c_{(n+(-1))}·( 1/(n^{2}) )·x^{n} )

ecuacions diferencials

d_{x}[y(x)]^{2} = ( n/(n+(-1)) )·y(x)·d_{xx}^{2}[y(x)]

y(x) = x^{n}

d_{x}[y(x)]^{2} = ( 1/(1+(-n)) )·y(x)·d_{xx}^{2}[y(x)]

y(x) = x^{(1/n)}

d_{x}[y(x)]^{2} = y(x)·d_{xx}^{2}[y(x)]

y(x) = e^{ax}

d_{x}[y(x)]^{3} = a·( y(x) )^{2}·d_{xx}^{2}[y(x)]

y(x) = e^{ax}

ecuació diferencial

d_{xx}^{2}[y(x)] = (1/x)·y(x)

y(x) = ∑ ( (1/n)·(1/(n+1)) )!·x^{(n+1)}

x·d_{xx}^{2}[ ∑ ( (1/n)·(1/(n+1)) )!·x^{(n+1)} ] = ∑ ( (1/(n+(-1)))·(1/n) )!·x^{n}

n=p+1

y(x) = ∑ ( (1/p)·(1/(p+1)) )!·x^{(p+1)}

ecuació diferencial series

x^{m+1}·d_{x...(m)...x}^{m}[y(x)] = y(x)


y(x) = ∑ ( (1/k!)·(k+m)!·x^{((k+1)+m)} )


x^{m+1}·d_{x...(m)...x}^{m}[ ∑ ( (1/k!)·(k+m)!·x^{((k+1)+m)} ) ] = ...
... ∑ ( (1/(k+1)!)·((k+1)+m)!·x^{((k+2)+m)} ) = ...
... ∑ ( (1/p!)·(p+m)!·x^{((p+1)+m)} ) & p = k+1

ecuació diferencial series

d_{x...(m)...x}^{m}[y(x)] = x^{n}·y(x)


y(x) = ∑ ( 1/( ((n+m)·k+(n+1))...(m)...((n+m)·k+(n+m)) )! )·x^{(n+m)·(k+1)} )


d_{x...(m)...x}^{m}[ ∑ ( 1/( ((n+m)·k+(n+1))...(m)...((n+m)·k+(n+m)) )! )·x^{(n+m)·(k+1)} ) ] = ...
... ∑ ( 1/( ((n+m)·(k+(-1))+(n+1))...(m)...((n+m)·(k+(-1))+(n+m)) )! )·x^{(n+m)·k+n} ) = ...
... ∑ ( 1/( ((n+m)·p+(n+1))...(m)...((n+m)·p+(n+m)) )! )·x^{(n+m)·(p+1)+n} ) & p = k+(-1)

ecuació diferencial series

x^{2m}·d_{x...(m)...x}^{m}[y(x)] = y(x)


y(x) = ∑ ( k!·x^{k+m} )


x^{2m}·d_{x...(m)...x}^{m}[ ∑ ( k!·x^{k+m} ) ] = ∑ ( (k+m)!·x^{(k+m)+m} )


p = k+m <==> ∑ ( (k+m)!·x^{(k+m)+m} ) = ∑ ( p!·x^{p+m} )

ecuació diferencial de tercer ordre

d_{x}[f(x)]·d_{xx}^{2}[f(x)]·d_{xxx}^{3}[f(x)] = g(x)


(1/2)·d_{x}[f(x)]^{2} [o(x)o] d_{xx}^{2}[f(x)] = ∫ [g(x)] d[x]


(1/6)·d_{x}[f(x)]^{([o(x)o] 2) [o(+)o] 1} = ∫ ∫ [g(x)] d[x]d[x]


f(x) = ∫ ( 6·∫ ∫ [g(x)] d[x]d[x] )^{(1/(([o(x)o] 2) [o(+)o] 1))} d[x]


d_{x}[f(x)]·d_{xx}^{2}[f(x)]·d_{xxx}^{3}[f(x)]·d_{xxxx}^{4}[f(x)] = g(x)


(1/2)·d_{x}[f(x)]^{2} [o(x)o] d_{xx}^{2}[f(x)] [o(x)o] d_{xxx}^{3}[f(x)] = ∫ [g(x)] d[x]


(1/6)·d_{x}[f(x)]^{([o(x)o] 2) [o(+)o] 1} [o(x)o]^{2} d_{xx}^{2}[f(x)] = ∫ ∫ [g(x)] d[x]d[x]


(1/24)·d_{x}[f(x)]^{(([o(x)o] 2) [o(+)o] 1) [o(+)o]^{2} 1} = ∫ ∫ ∫ [g(x)] d[x]d[x]d[x]


f(x) = ∫ ( 24·∫ ∫ ∫ [g(x)] d[x]d[x]d[x] )^{(1/((([o(x)o] 2) [o(+)o] 1) [o(+)o]^{2} 1))} d[x]


∫ ( f(x) )^{n} [o(x)o] d_{x}[f(x)] d[x] = ( 1/(n+1) )·( f(x) )^{([o(x)o] n) [o(+)o] 1}
∫ ( f(x) )^{n} [o(x)o]^{p} d_{x}[f(x)] d[x] = ( 1/(n+1) )·( f(x) )^{([o(x)o] n) [o(+)o]^{p} 1}

ecuació diferencial exponencial integral

d_{xx}^{2}[ e^{[o(x)o]^{2} ∫ [ ( ∫ [2·g(x)] d[x] )^{(1/2)} ] d[x] } ] = ...
... g(x)·e^{[o(x)o]^{2} ∫ [ ( ∫ [2·g(x)] d[x] )^{(1/2)} ] d[x] }


d_{xx}^{2}[ e^{[o(x)o]^{2}f(x)} ] = e^{[o(x)o]^{2}f(x)}·d_{x}[f(x)]·d_{xx}^{2}[f(x)]


d_{xx}^{2}[y(x)] = g(x)·y(x)


y(x) = e^{[o(x)o]^{2} ∫ [ ( ∫ [2·g(x)] d[x] )^{(1/2)} ] d[x] }


d_{x}[ e^{[o(x)o]^{n}f(x)} ] = e^{[o(x)o]^{n}f(x)} [o(x)o]^{n+(-1)} d_{x}[f(x)]


d_{xx}^{2}[ e^{[o(x)o]^{n}f(x)} ] = ...
... e^{[o(x)o]^{n}f(x)} [o(x)o]^{n+(-2)} d_{x}[f(x)] [o(x)o]^{n+(-2)} d_{xx}^{2}[f(x)]


d_{x,...,x}^{n}[ e^{[o(x)o]^{n}f(x)} ] = e^{[o(x)o]^{n}f(x)}·d_{x}[f(x)]·...(n)...·d_{x,...,x}^{n}[f(x)]

ecuació diferencial series

y(x) = ∑ ( k^{m}·x^{(k+1)} )


x·d_{x}[y(x)] = ∑ ( k^{(m+1)}·x^{(k+1)} )+y(x)


x·d_{x}[ ∑ ( k^{m}·x^{(k+1)} ) ] = ∑ ( (k+1)·k^{m}·x^{(k+1)} )

ecuació diferencial series

y(x) = ∑ ( k!·x^{(k+1)} )


x^{2}·d_{x}[y(x)] = y(x)


x^{2}·d_{x}[ ∑ ( k!·x^{(k+1)} ) ] = ∑ ( (k+1)!·x^{(k+2)} )


p = k+1 <==> p+1 = k+2


∑ ( (k+1)!·x^{(k+2)} ) = ∑ ( p!·x^{(p+1)} )

operadors diferencials amb coeficients constants

d_{x}[y(x)]+a·y(x) = 0
y(x) = e^{(-a)x}


d_{xx}[y(x)]+(a+b)·d_{x}[y(x)]+(ab)·y(x) = 0
( d_{x}[...]+a )o( d_{x}[...]+b )[y(x)] = 0
y(x) = e^{(-a)x}+e^{(-b)x}


x·d_{x}[y(x)]+a·y(x) = 0
y(x) = x^{(-a)}


x·d_{x}[ x·d_{x}[y(x)] ]+(a+b)·x·d_{x}[y(x)]+(ab)·y(x) = 0
( x·d_{x}[...]+a )o( x·d_{x}[...]+b )[y(x)] = 0
y(x) = x^{(-a)}+x^{(-b)}


x^{(p+1)}·d_{x}[y(x)]+a·y(x) = 0
y(x) = e^{(a/p)·(1/x^{p})}


x^{(p+1)}·d_{x}[ x^{(p+1)}·d_{x}[y(x)] ]+(a+b)·x^{(p+1)}·d_{x}[y(x)]+(ab)·y(x) = 0
( x^{(p+1)}·d_{x}[...]+a )o( x^{(p+1)}·d_{x}[...]+b )[y(x)] = 0
y(x) = e^{(a/p)·(1/x^{p})}+e^{(b/p)·(1/x^{p})}


x^{((-p)+1)}·d_{x}[y(x)]+a·y(x) = 0
y(x) = e^{(-1)·(a/p)·x^{p}}


x^{((-p)+1)}·d_{x}[ x^{((-p)+1)}·d_{x}[y(x)] ]+(a+b)·x^{((-p)+1)}·d_{x}[y(x)]+(ab)·y(x) = 0
( x^{((-p)+1)}·d_{x}[...]+a )o( x^{((-p)+1)}·d_{x}[...]+b )[y(x)] = 0
y(x) = e^{(-1)·(a/p)·x^{p}}+e^{(-1)·(b/p)·x^{p}}

lagranià para-magnétic eléctric

d_{tt}^{2}[x]= ( d_{t}[x]^{n} )


d_{tt}[x(t)] = a^{(1/2)+(-1)(n/4)}·...
... ( a^{(-1)(1+(-n))/4)} )^{n/(1+(-n))}·a^{(-1)(1+(-n))/2)} = 1


( x(t) )^{n} = a^{(n/2)+(-1)(n^{2}/4)}·...
... ( a^{(-1)(1+(-n))(2+(-n))/4} )^{n/(1+(-n))} = 1


( x(t) ) = a^{(1/2)+(-1)(n/4)}·...
... ( a^{(-1)(1+(-n))/4} )^{(2+(-n))/(1+(-n))}·t^{( (2+(-n))/(1+(-n)) )}


d_{tt}[x(t)] = a^{(1/4)+(-1)(n/4)}·...
... ( a^{(-1)(1+(-n))/4) )^{n/(1+(-n))}·a^{(-1)(1+(-n))/2)} = a^{(-1)(1/4)}


( x(t) )^{n} = a^{(n/4)+(-1)(n^{2}/4)}·...
... ( a^{(-1)(1+(-n))(2+(-n))/4} )^{n/(1+(-n))} = a^{(-1)(n/4)}


( x(t) ) = a^{(-1)(1/4)+(-1)(n/4)}·...
... ( a^{(-1)(1+(-n))/4} )^{(2+(-n))/(1+(-n))}·t^{(2+(-n))/(1+(-n))}


d_{tt}^{2}[x]= ( a^{(n+(-1))(1/4))}·d_{t}[x]^{n}/c^{n} )
m·d_{tt}^{2}[x]= (k_{e}·pq)·d_{t}[x]^{n}/c^{n} )


a^{(n+(-1))(1/4))} = ( (k_{e}·pq)/(mc^{n}) )


a = ( (k_{e}·pq)/(mc^{n}) )^{( 1/(n+(-1))(1/4)) )}

lagranià para-eléctric

d_{tt}^{2}[x]= ( x^{n}/t^{n} )


d_{tt}[x(t)] = a^{(1/2)+(-1)(n/4)}·...
... ( a^{(-1)(1+(-n))/4)} )^{n/(1+(-n))}·a^{(-1)(1+(-n))/2)} = 1


( x(t) )^{n} = a^{(n/2)+(-1)(n^{2}/4)}·...
... ( a^{(-1)(1+(-n))(2+(-n))/4} )^{n/(1+(-n))} = 1


( x(t) ) = a^{(1/2)+(-1)(n/4)}·...
... ( a^{(-1)(1+(-n))/4} )^{(2+(-n))/(1+(-n))}·t^{(2+(-n))/(1+(-n))}


d_{tt}[x(t)] = a^{(1/4)+(-1)(n/4)}·...
... ( a^{(-1)(1+(-n))/4) )^{n/(1+(-n))}·a^{(-1)(1+(-n))/2)} = a^{(-1)(1/4)}


( x(t) )^{n} = a^{(n/4)+(-1)(n^{2}/4)}·...
... ( a^{(-1)(1+(-n))(2+(-n))/4} )^{n/(1+(-n))} = a^{(-1)(n/4)}


( x(t) ) = a^{(-1)(1/4)+(-1)(n/4)}·...
... ( a^{(-1)(1+(-n))/4} )^{(2+(-n))/(1+(-n))}·t^{(2+(-n))/(1+(-n))}


d_{tt}^{2}[x]= ( a^{(n+(-1))(1/4))}·x^{n}/t^{n} )
m·d_{tt}^{2}[x]= (k_{e}·pq)·x^{n}/(ct)^{n} )


a^{(n+(-1))(1/4))} = ( (k_{e}·pq)/(mc^{n}) )


a = ( (k_{e}·pq)/(mc^{n}) )^{( 1/(n+(-1))(1/4)) )}

ecuacions diferencials: binomi

d_{t}[x]^{n}+d_{t}[y]^{n} = (x+y)^{2}


x = ( ( 2^{(1/n)}·(n+(-2))/n )·t )^{( n/(n+(-2)) )}
y = ( ( 2^{(1/n)}·(n+(-2))/n )·t )^{( n/(n+(-2)) )}


d_{t}[x]^{n}+d_{t}[y]^{n} = (x+y)^{3}


x = ( ( 2^{(2/n)}·(n+(-3))/n )·t )^{( n/(n+(-3)) )}
y = ( ( 2^{(2/n)}·(n+(-3))/n )·t )^{( n/(n+(-3)) )}


d_{t}[x]^{n}+d_{t}[y]^{n} = (x+y)^{m}


x = ( ( 2^{((m+(-1))/n)}·(n+(-m))/n )·t )^{( n/(n+(-m)) )}
y = ( ( 2^{((m+(-1))/n)}·(n+(-m))/n )·t )^{( n/(n+(-m)) )}


d_{t}[x]^{n}+d_{t}[y]^{n} = (x+y)^{(1/m)}


x = ( ( 2^{((1/m)+(-1))}·(n+(-1)(1/m))/n )·t )^{( n/(n+(-1)(1/m)) )}
y = ( ( 2^{((1/m)+(-1))}·(n+(-1)(1/m))/n )·t )^{( n/(n+(-1)(1/m)) )}

índex de matemàtiques

index de etiquetes matemàtiques

matemáticas: modelo de la equación diferencial de primer orden de logaritmo no inverso

f(x) = e^{(m/x^{n})}
ln( f(x) ) = (m/x^{n})


d_{z}[h(z)] + (-1)·( x^{n+1}/m )·h(z) = 0


h(z) = e^{(x^{n+1}/m)·z}


( ln(e^{(x^{n+1}/m)·(m/x^{n})}) ) = x


( ln(e^{(x^{n+1}/m)·ln( f(x) )}) ) = x


( ln( h( ln(f(x)) ) ) ) = x

matemáticas: modelo enésimo de la ecuación diferencial de primer orden

f(x) = e^{(m/x^{n})}
ln( f(x) ) = (m/x^{n})


d_{y}[g(y)] + (-1)·( m/x^{n+1} )·g(y) = 0


g(y) = e^{(m/x^{n+1})·y}


( 1/ln(e^{(m/x^{n+1})·(x^{n}/m)}) ) = x


( 1/ln(e^{(m/x^{n+1})·( 1/ln( f(x) ) )}) ) = x


( 1/ln( g( 1/ln(f(x)) ) ) ) = x

ecuació diferencial potencia integral elíptica

ecuació diferencial elíptica:
d_{xx}^{2}[y(x)] + k·d_{x}[y(x)]^{n} = (-1)^{(-1)/(n+(-1))}·k·( cotan( (n+(-1))k·x ) )^{(n+(-2))/(n+(-1))}


y(x) = ∫ [ ( (-1)·tan( (n+(-1))k ) )^{1/(n+(-1))} ] d[x]


(-1)·k·( (-1)·cotan( (n+(-1))k ) )^{(-n)/(n+(-1))}·( 1+( cotan( (n+(-1))k·x ) )^{2}) +...
... k·( (-1)·tan( (n+(-1))k ) )^{n/(n+(-1))} = (-1)^{(-1)/(n+(-1))}·k·( cotan( (n+(-1))k·x ) )^{(n+(-2))/(n+(-1))}


d_{xx}^{2}[y(x)] + k·d_{x}[y(x)]^{n} = (-1)·k·( tan( (n+(-1))k·x ) )^{(n+(-2))/(n+(-1))}


y(x) = ∫ [ ( cotan( (n+(-1))k ) )^{1/(n+(-1))} ] d[x]


(-1)·k·( tan( (n+(-1))k ) )^{(-n)/(n+(-1))}( 1+( tan( (n+(-1))k·x ) )^{2}) +...
... k·( cotan( (n+(-1))k ) )^{n/(n+(-1))} = (-1)·k·( tan( (n+(-1))k·x ) )^{(n+(-2))/(n+(-1))}


elíptiques sin(x):
f(x) = ( sin( (n+(-1))k·x ) )^{1/(n+(-1))}


( ∫ [ sin( (n+(-1))k·x ) ] d[x]·(n+(-1))·k )^{(-1)/(n+(-1))}·d_{x}[( sin( (n+(-1))k·x ) )^{1/(n+(-1))}]


∫ [ ( (-1)·cos( (n+(-1))k·x ) )^{(-1)/(n+(-1))}·d_{x}[( sin( (n+(-1))k·x ) )^{1/(n+(-1))}] ] d[x] =...
...( (-1)·sin( (n+(-1))k·x ) )^{[o(x)o](-1)/(n+(-1))} [o(x)o] ...
...( (-1)·cos( (n+(-1))k·x ) )^{[o(x)o]((-n)+2)/(n+(-1))} [o(x)o] ...
... sin( (n+(-1))k·x ) [o(x)o] kx = ..


... (-1)·k·( tan[o(x)o]( (n+(-1))k·x ) )^{[o(x)o](n+(-2))/(n+(-1))} = ...
... (-1)^{(-1)(n+(-1))/(n+(-1))}·k·( tan[o(x)o]( (n+(-1))k·x ) )^{[o(x)o](n+(-2))/(n+(-1))} = ...
... (-1)^{((-n)+1)/(n+(-1))}·k· ∫ [ ( (-1)·cotan( (n+(-1))k·x ) )^{(n+(-2))/(n+(-1))} ] d[x] = ...
... (-1)^{(-1)/(n+(-1))}·k· ∫ [ ( cotan( (n+(-1))k·x ) )^{(n+(-2))/(n+(-1))} ] d[x]


elíptiques cos(x):
f(x) = ( cos( (n+(-1))k·x ) )^{1/(n+(-1))}


( ∫ [ cos( (n+(-1))k·x ) ] d[x]·(n+(-1))·k )^{(-1)/(n+(-1))}·d_{x}[( cos( (n+(-1))k·x ) )^{1/(n+(-1))}]


∫ [ ( sin( (n+(-1))k·x ) )^{(-1)/(n+(-1))}·d_{x}[( cos( (n+(-1))k·x ) )^{1/(n+(-1))}] ] d[x] =...
...( (-1)·cos( (n+(-1))k·x ) )^{[o(x)o](-1)/(n+(-1))} [o(x)o] ...
...( sin( (n+(-1))k·x ) )^{[o(x)o]((-n)+2)/(n+(-1))} [o(x)o] ...
... cos( (n+(-1))k·x ) [o(x)o] kx = ..


... (-1)^{(-1)/(n+(-1))}·k·( cotan[o(x)o]( (n+(-1))k·x ) )^{[o(x)o](n+(-2))/(n+(-1))} =...
... (-1)^{(-1)/(n+(-1))}·k· ∫ [ ( (-1)^{(-1)}·tan( (n+(-1))k·x ) )^{(n+(-2))/(n+(-1))} ] d[x] =...
... (-1)·k· ∫ [ ( tan( (n+(-1))k·x ) )^{(n+(-2))/(n+(-1))} ] d[x]

ecuació diferencial potencia integral hiperbólica

hiperbóliques sinh(x):
f(x) = ( sinh( (n+(-1))k·x ) )^{1/(n+(-1))}


( ∫ [ sinh( (n+(-1))k·x ) ] d[x]·(n+(-1))·k )^{(-1)/(n+(-1))}·d_{x}[( sinh( (n+(-1))k·x ) )^{1/(n+(-1))}]


∫ [ ( cosh( (n+(-1))k·x ) )^{(-1)/(n+(-1))}·d_{x}[( sinh( (n+(-1))k·x ) )^{1/(n+(-1))}] ] d[x] =...
...( sinh( (n+(-1))k·x ) )^{[o(x)o](-1)/(n+(-1))} [o(x)o] ...
...( cosh( (n+(-1))k·x ) )^{[o(x)o]((-n)+2)/(n+(-1))} [o(x)o] ...
... sinh( (n+(-1))k·x ) [o(x)o] kx = ..


... k·( tanh[o(x)o]( (n+(-1))k·x ) )^{[o(x)o](n+(-2))/(n+(-1))} = ...
... k· ∫ [ ( cotanh( (n+(-1))k·x ) )^{(n+(-2))/(n+(-1))} ] d[x]


hiperbóliques cosh(x):
f(x) = ( cosh( (n+(-1))k·x ) )^{1/(n+(-1))}


( ∫ [ cosh( (n+(-1))k·x ) ] d[x]·(n+(-1))·k )^{(-1)/(n+(-1))}·d_{x}[( cosh( (n+(-1))k·x ) )^{1/(n+(-1))}]


∫ [ ( sinh( (n+(-1))k·x ) )^{(-1)/(n+(-1))}·d_{x}[( cosh( (n+(-1))k·x ) )^{1/(n+(-1))}] ] d[x] =...
...( cosh( (n+(-1))k·x ) )^{[o(x)o](-1)/(n+(-1))} [o(x)o] ...
...( sinh( (n+(-1))k·x ) )^{[o(x)o]((-n)+2)/(n+(-1))} [o(x)o] ...
... cosh( (n+(-1))k·x ) [o(x)o] kx = ..


... k·( cotanh[o(x)o]( (n+(-1))k·x ) )^{[o(x)o](n+(-2))/(n+(-1))} =...
... k· ∫ [ ( tanh( (n+(-1))k·x ) )^{(n+(-2))/(n+(-1))} ] d[x]


ecuació diferencial hiperbólica:
d_{xx}^{2}[y(x)] + k·d_{x}[y(x)]^{n} = k·( cotanh( (n+(-1))k·x ) )^{(n+(-2))/(n+(-1))}


y(x) = ∫ [ ( tanh( (n+(-1))k ) )^{1/(n+(-1))} ] d[x]


(-1)·k·( cotanh( (n+(-1))k ) )^{(-n)/(n+(-1))}·( 1+(-1)·( cotanh( (n+(-1))k·x ) )^{2}) +...
... k·( tanh( (n+(-1))k ) )^{n/(n+(-1))} = k·( cotanh( (n+(-1))k·x ) )^{(n+(-2))/(n+(-1))}


d_{xx}^{2}[y(x)] + k·d_{x}[y(x)]^{n} = k·( tanh( (n+(-1))k·x ) )^{(n+(-2))/(n+(-1))}


y(x) = ∫ [ ( cotanh( (n+(-1))k ) )^{1/(n+(-1))} ] d[x]


(-1)·k·( tanh( (n+(-1))k ) )^{(-n)/(n+(-1))}( 1+(-1)·( tanh( (n+(-1))k·x ) )^{2}) +...
... k·( cotanh( (n+(-1))k ) )^{n/(n+(-1))} = k·( tanh( (n+(-1))k·x ) )^{(n+(-2))/(n+(-1))}