restart:with(plots):with(DETools):
EDO'S LINEARES DE ORDEM MAIS ALTA (COM COEFICIENTES CONSTANTES)
Come\303\247amos com equa\303\247\303\265es homog\303\252neas .
Aqui est\303\241 um exemplo tirado do livro:
edo:=diff(y(t),t,t,t,t)+diff(y(t),t,t,t)-7*diff(y(t),t,t)-diff(y(t),t)+6*y(t)=0;
LywsLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUYlNiQtRiU2JC1JInlHNiI2I0kidEdGMEYyRjJGMkYyIiIiRihGMyomIiIoRjNGKkYzISIiRixGNiomIiInRjNGLkYzRjMiIiE=
A solu\303\247\303\243o geral \303\251:
dsolve(edo);
Ly1JInlHNiI2I0kidEdGJSwqKiZJJF9DMUdGJSIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJUYmRitGKyomSSRfQzJHRiVGKy1GLTYjLCQqJiIiI0YrRidGK0YrRitGKyomSSRfQzNHRiVGKy1GLTYjLCQqJiIiJEYrRidGKyEiIkYrRisqJkkkX0M0R0YlRistRi02IywkRidGP0YrRis=
Vamos confirmar que a solu\303\247\303\243o encontrada pelo Maple est\303\241 de acordo com a teoria. Os n\303\272meros que aparecem multiplicando t nos expoentes devem ser ra\303\255zes (neste caso, simples) da equa\303\247\303\243o caracter\303\255stica. Isto pode ser confirmado olhando o gr\303\241fico do polin\303\264mio caracter\303\255stico:
plot(r^4+r^3-7*r^2-r+6, r=-3.5..3);
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
Agora, vamos resover o seguinte problema de valor inicial:
cis:=y(0)=1, D(y)(0)=0, D(D(y))(0)=-2, D(D((D(y))))(0)=-1;
NiYvLUkieUc2IjYjIiIhIiIiLy0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGJUYnRigvLS0tSSNAQEdGLjYkRi0iIiNGMUYnISIjLy0tLUY2NiRGLSIiJEYxRichIiI=
dsolve({edo,cis});
Ly1JInlHNiI2I0kidEdGJSwqKiYjIiM2IiIpIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlRiZGLUYtKiYjIiIjIiIkRi0tRi82IywkKiZGNUYtRidGLUYtRi0hIiIqJiNGLUYsRi0tRi82IywkKiZGNkYtRidGLUY7Ri1GOyomIyIiJiIjN0YtLUYvNiMsJEYnRjtGLUYt
sol:=rhs(%):
plot(sol,t=0..1.1);
LSUlUExPVEc2Jy0lJ0NVUlZFU0c2JDdTNyQkIiIhISIiJCIjNSEiIjckJCIyTUxMJGVwbyhSIyEjPSQiMV8iNE9eRFUqKiohIzs3JCQiMW5tVGc6IVJbJSEjPCQiMTR0KTReNyh6KiohIzs3JCQiMU1MJDMybyskbyEjPCQiMVYyRjF6a18qKiEjOzckJCIxTkwkZVwneSI+KiEjPCQiMUUpWy8pZmw4KiohIzs3JCQiMm5tVCYzIUdVOiIhIzwkIjFvIypmVzEjRycpKiEjOzckJCIyTUwzRiYpW0BQIiEjPCQiMSgqZVQkPlBZISkqISM7NyQkIi52VkUkeihmIiEjOCQiMS9XUHlZIUd0KiEjOzckJCIyTkwzeC5iNiQ9ISM8JCIxMXcsKEdkY2sqISM7NyQkIi52b1RvUDEjISM4JCIxMz1bJG9dX2EqISM7NyQkIjFubTtwKVJJSSMhIzskIjF6STVleUNGJSohIzs3JCQiMk5MJGUlSCF6OEQhIzwkIjFZaWtPQlI1JCohIzs3JCQiMjArK3ZOWDV2IyEjPCQiMSRHeU4xJW9qIiohIzs3JCQiLERodSMqKUghIzYkIjAieWh1IVElKioqKSEjOjckJCIsRGtfKT1LISM2JCIxXEJFP3BGQykpISM7NyQkIjJMTDN4W0p0VSQhIzwkIjFDOzNmKVEsbCkhIzs3JCQiMW5tOyoqSEJ2TyEjOyQiMVZDeUFXVUIlKSEjOzckJCIxbm1tJzRRXylRISM7JCIxLEAzRzciUUApISM7NyQkIi52JFxSX0hUISM4JCIxLmlFZyY0J1t6ISM7NyQkIjFubW1AJGVkTSUhIzskIjB4VCMpKjNuJHAoISM6NyQkIjIxK3YkKnAsSWUlISM8JCIxcy55WSdSNFIoISM7NyQkIjEsXTcpXDgqM1shIzskIjEqM2NSOGchenEhIzs3JCQiMW5tVGcoR1kvJiEjOyQiMXQkUjpkO3hzJyEjOzckJCIxbjthYCopM2hfISM7JCIxNlQ2eV5aIVEnISM7NyQkIjFMTGU5MGQlXCYhIzskIjFXISp5Sl0xeWYhIzs3JCQiMUwkM3hCIzRQZCEjOyQiMUZjImUqKVJ4XyYhIzs3JCQiMSxdaTUiMyNbZiEjOyQiMWZ0QXppVDJeISM7NyQkIjFNTDNQIT5pPCchIzskIjEnKiplKUdJQUFZISM7NyQkIitObXg2ayEjNSQiMVQlR3JNYF0zJSEjOzckJCItdil5QkFrJyEjNyQiMUJIeHd0QkFOISM7NyQkIjEsXVBmSz5sbyEjOyQiMUB0KydRQTElSCEjOzckJCItRFRHdzdyISM3JCIyV21tZSp5c1xBISM8NyQkIjFtbW1AXkBOdCEjOyQiMmMvXUIiPj4nZSIhIzw3JCQiMSwrXTcvdHN2ISM7JCIwVTlIRmxhSSkhIzs3JCQiMU0kM3hjYXp5KCEjOyQiMjx6ZToiKSp5OTUhIz03JCQiMSwrXTw5REIhKSEjOyQhMVJSVDZ0ZGt1ISM8NyQkIjFtOy87dWtXIykhIzskITIlKjQpWyUpR1MmZiIhIzw3JCQiMSxdKG8tcWdaKSEjOyQhMHBDOFhNJ1FEISM6NyQkIjFubTtIekstKCkhIzskITJGIUh6KjRUKj1OISM8NyQkIi52VikqKT5SKikhIzgkITFkU2I0ODs1WSEjOzckJCIxTkxMalJMbiIqISM7JCExJ3BQPF87dHMmISM7NyQkIjFOTGVIXGorJSohIzskITF3ZG1zPWVTcCEjOzckJCIxbzsvWVMrSycqISM7JCExKSpvJHBAQiQ9IykhIzs3JCQiMSwrK04jM1klKSohIzskITFWZid5KSopR2glKiEjOzckJCIyb20ieml3IykzNSEjOyQhMlBdeig0cjcoNCIhIzs3JCQiMk5MTFZsQDEuIiEjOyQhMHpLaS41LkMiISM5NyQkIjIsXVBcZmVRMCIhIzskITJkSSQ0QXEoPVMiISM7NyQkIjItXWk/Sio0dzUhIzskITA4YWxZbWJjIiEjOTckJCIjNiEiIiQhMk8tOVdcaD12IiEjOy0lJkNPTE9SRzYmJSRSR0JHJCIjNSEiIiQiIiEhIiIkIiIhISIiLSUlVklFV0c2JDskIiIhISIiJCIjNiEiIiUoREVGQVVMVEctJSpBWEVTU1RZTEVHNiMlJ05PUk1BTEctJShTQ0FMSU5HRzYjJS5VTkNPTlNUUkFJTkVERy0lJVJPT1RHNictJSlCT1VORFNfWEc2IyQiJDUlISIiLSUpQk9VTkRTX1lHNiMkIiQ/IiEiIi0lLUJPVU5EU19XSURUSEc2IyQiJXFaISIiLSUuQk9VTkRTX0hFSUdIVEc2IyQiJV1JISIiLSUpQ0hJTERSRU5HNiI=
Agora, outra equa\303\247\303\243o (tamb\303\251m tirada do livro):
edo:=diff(y(t),t,t,t,t)-y(t)=0;
LywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUYlNiQtRiU2JC1JInlHNiI2I0kidEdGMEYyRjJGMkYyIiIiRi4hIiIiIiE=
dsolve(edo);
Ly1JInlHNiI2I0kidEdGJSwqKiZJJF9DMUdGJSIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJUYmRitGKyomSSRfQzJHRiVGKy1GLTYjLCRGJyEiIkYrRisqJkkkX0MzR0YlRistSSRzaW5HRi5GJkYrRisqJkkkX0M0R0YlRistSSRjb3NHRi5GJkYrRis=
De novo, vamos comparar a forma da equa\303\247\303\243o geral com o gr\303\241fico do polin\303\264mio caracter\303\255stico:
plot(r^4-1, r=-1.2..1.2);
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
Como s\303\263 2 ra\303\255zes reais: -1 e 1. Al\303\251m disso, como elas n\303\243o tangenciam o eixo horizontal, s\303\243o ambas ra\303\255zes simples. Como o polin\303\264mio tem grau 4, devem existir 2 ra\303\255zes complexas conjugadas.
Agora, camos resolver a EDO para duas condi\303\247\303\265es iniciais:
cis1:=y(0)=7/2, D(y)(0)=-4, D(D(y))(0)=5/2, D(D((D(y))))(0)=-2;
NiYvLUkieUc2IjYjIiIhIyIiKCIiIy8tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjRiVGJyEiJS8tLS1JI0BAR0YwNiRGL0YrRjNGJyMiIiZGKy8tLS1GOTYkRi8iIiRGM0YnISIj
dsolve({edo,cis1});
Ly1JInlHNiI2I0kidEdGJSwoKiYiIiQiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkRichIiJGK0YrLUkkc2luR0YuRiZGMyomI0YrIiIjRistSSRjb3NHRi5GJkYrRis=
sol1:=rhs(%):
plot(sol1,t=0..15);
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
cis2:=y(0)=7/2, D(y)(0)=-4, D(D(y))(0)=5/2, D(D((D(y))))(0)=-15/8;
NiYvLUkieUc2IjYjIiIhIyIiKCIiIy8tLUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjRiVGJyEiJS8tLS1JI0BAR0YwNiRGL0YrRjNGJyMiIiZGKy8tLS1GOTYkRi8iIiRGM0YnIyEjOiIiKQ==
dsolve({edo,cis2});
Ly1JInlHNiI2I0kidEdGJSwqKiYjIiIiIiNLRistSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiVGJkYrRisqJiMiIyYqRixGKy1GLjYjLCRGJyEiIkYrRisqJiMiIzwiIztGKy1JJHNpbkdGL0YmRitGOComI0YrIiIjRistSSRjb3NHRi9GJkYrRis=
sol2:=rhs(%):
Comparando as duas solu\303\247\303\265es: A segunda cont\303\251m um exp(t) e rapidamente explode; j\303\241 a primeira permanece limitada quando t aumenta.
plot([sol1,sol2],t=0..8,color=[red,blue]);
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
Mais um exemplo:
edo:=diff(y(t),t,t,t,t)-2*diff(y(t),t,t)+y(t)=0;
LywoLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUYlNiQtRiU2JC1JInlHNiI2I0kidEdGMEYyRjJGMkYyIiIiRiohIiNGLkYzIiIh
dsolve(edo);
Ly1JInlHNiI2I0kidEdGJSwqKiZJJF9DMUdGJSIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJUYmRitGKyooSSRfQzJHRiVGK0YsRitGJ0YrRisqJkkkX0MzR0YlRistRi02IywkRichIiJGK0YrKihJJF9DNEdGJUYrRjVGK0YnRitGKw==
Note que apareceram solu\303\247\303\265es da forma t*exp(const*t). Isso \303\251 porque a equa\303\247\303\243o carater\303\255stica tem ra\303\255zes n\303\243o-simples. Olhando o gr\303\241fico do polin\303\264mio caracter\303\255stico, podemos detectar as ra\303\255zes n\303\243o-simples como pontos de tang\303\252ncia com o eixo horizontal:
plot(r^4-2*r^2+1,r=-1.5..1.5);
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
Mais um exemplo:
edo:=diff(y(t),t,t,t,t)+y(t)=0;
LywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUYlNiQtRiU2JC1JInlHNiI2I0kidEdGMEYyRjJGMkYyIiIiRi5GMyIiIQ==
dsolve(edo);
Ly1JInlHNiI2I0kidEdGJSwqKihJJF9DMUdGJSIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCQqKCNGKyIiI0YrKUY1RjRGK0YnRishIiJGKy1JJHNpbkdGLjYjLCRGM0YrRitGNyooSSRfQzJHRiVGKy1GLUY6RitGOEYrRjcqKEkkX0MzR0YlRitGLEYrLUkkY29zR0YuRjpGK0YrKihJJF9DNEdGJUYrRj5GK0ZBRitGKw==
Note que a equa\303\247\303\243o caracter\303\255stica \303\251 r^4 + 1 =0, que n\303\243o tem ra\303\255zes reais.
Equa\303\247\303\265es n\303\243o-homog\303\252neas
Aqui est\303\241 uma equa\303\247\303\243o n\303\243o homog\303\252nea cuja parte linear apareceu num exemplo acima:
edo:=diff(y(t),t,t,t,t)+diff(y(t),t,t,t)-7*diff(y(t),t,t)-diff(y(t),t)+6*y(t)= 5*cos(3*t);
LywsLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUYlNiQtRiU2JC1JInlHNiI2I0kidEdGMEYyRjJGMkYyIiIiRihGM0YqISIoRiwhIiJGLiIiJywkLUkkY29zRzYkRiZJKF9zeXNsaWJHRjA2IywkRjIiIiQiIiY=
dsolve(edo);
Ly1JInlHNiI2I0kidEdGJSwuLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJEYnIiIkIyIiJiIkYyItSSRzaW5HRitGLiMhIiJGMyomSSRfQzFHRiUiIiItSSRleHBHRitGJkY6RjoqJkkkX0MyR0YlRjotRjw2IywkRichIiRGOkY6KiZJJF9DM0dGJUY6LUY8NiMsJEYnRjdGOkY6KiZJJF9DNEdGJUY6LUY8NiMsJEYnIiIjRjpGOg==
Como ter\303\255amos encontrado isso \303\240 m\303\243o? Basta encontrar a solu\303\247\303\243o particular.
Usamos o m\303\251todo dos coeficientes a determinar.
Aqui vai uma m\303\251todo, com a ajudinha do Maple:
O lado esquerdo da equa\303\247\303\243o diferencial pode ser pensado como um funcional linear:
L := y -> diff(y,t,t,t,t)+diff(y,t,t,t)-7*diff(y,t,t)-diff(y,t)+6*y;
Zio2I0kieUc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCwtSSVkaWZmRyUqcHJvdGVjdGVkRzYnOSRJInRHRiVGL0YvRi8iIiItRis2JkYuRi9GL0YvRjAtRis2JUYuRi9GLyEiKC1GKzYkRi5GLyEiIkYuIiInRiVGJUYl
L recebe como entrada uma fun\303\247\303\243o de t e devolve outra fun\303\247\303\243o de t. (Isso \303\251 chamado de "funcional".) Exemplos:
L(exp(2*t)*cos(t));
LCYqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInRHRikiIiMiIiItSSRjb3NHRiY2I0YsRi4hI0EqJkYkRi4tSSRzaW5HRiZGMUYuISIn
L(1/t);
LCwqJEkidEc2IiEiJiIjQyokRiQhIiUhIicqJEYkISIkISM5KiRGJCEiIyIiIiokRiQhIiIiIic=
Dizemos que o funcional L \303\251 linear porque ele satisfaz o princ\303\255pio da superposi\303\247\303\243o L(f+g) = L(f) + L(g). Vamos verificar isso com um exemplo:
L(exp(2*t)*cos(t) + 1/t);
LDAqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInRHRikiIiMiIiItSSRjb3NHRiY2I0YsRi4hI0EqJkYkRi4tSSRzaW5HRiZGMUYuISInKiRGLCEiJiIjQyokRiwhIiVGNiokRiwhIiQhIzkqJEYsISIjRi4qJEYsISIiIiIn
Tamb\303\251m vale que L(c*f) = c*L(f) se c \303\251 uma constante. Exemplo:
L(1000*exp(2*t)*cos(t));
LCYqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCRJInRHRikiIiMiIiItSSRjb3NHRiY2I0YsRi4hJis/IyomRiRGLi1JJHNpbkdGJkYxRi4hJStn
Agora, voltemos ao problema de resolver a equa\303\247\303\243o homog\303\252nea l\303\241 de cima, ou seja, L(y) = 5*cos(3*t).
Resolveremos isso com uma sequencia
Vejamos o que L faz com cossenos:
L(cos(omega*t));
LCwqJi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKiZJJm9tZWdhR0YpIiIiSSJ0R0YpRi1GLUYsIiIlRi0qJi1JJHNpbkdGJkYqRi1GLCIiJEYtKiZGJEYtRiwiIiMiIigqJkYxRi1GLEYtRi1GJCIiJw==
E com senos:
L(sin(omega*t));
LCwqJi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKiZJJm9tZWdhR0YpIiIiSSJ0R0YpRi1GLUYsIiIlRi0qJi1JJGNvc0dGJkYqRi1GLCIiJCEiIiomRiRGLUYsIiIjIiIoKiZGMUYtRixGLUY0RiQiIic=
Em ambos os casos, a resposta de L foi uma mistura de senos com cossenos, mas a frequ\303\252ncia angular omega foi preservada. Como queremos que a sa\303\255da envolva cos(3*t), parece boa id\303\251ia escolher omega=3.
L(cos(3*t));
LCYtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoIiIkIiRdIi1JJHNpbkdGJUYpIiNJ
L(sin(3*t));
LCYtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoIiIkIiRdIi1JJGNvc0dGJUYpISNJ
Nenhum dos dois d\303\241 a resposta desejada. Mas, se fizermos uma combina\303\247\303\243o linear:
L(A*cos(3*t)+B*sin(3*t));
LCoqJkkiQUc2IiIiIi1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCRJInRHRiUiIiRGJiIkXSIqJkkiQkdGJUYmLUkkc2luR0YpRixGJkYwKiZGJEYmRjNGJiIjSSomRjJGJkYnRiYhI0k=
O coeficiente do cos(3*t) \303\251 150A - 30B, que queremos que seja 5; o coeficiente de sin(3*t) \303\251 150B + 30A, que queremos que seja 0. Portanto podemos encontrar A e B resolvendo um sistema linear.
solve({150*A - 30*B=5,150*B+30*A=0});
PCQvSSJBRzYiIyIiJiIkYyIvSSJCR0YlIyEiIkYo
Isso nos d\303\241 a solu\303\247\303\243o particular desejada:
L(5/156*cos(3*t)-1/156*sin(3*t));
LCQtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkSSJ0R0YoIiIkIiIm
O m\303\251todo dos coeficientes a determinar consiste em encontrar um bom chute do formato da solu\303\247\303\243o particular, e depois encontrar os coeficientes resolvendo um sistema linear.
Vejamos mais um exemplo: Encontrar uma solu\303\247\303\243o particular da equa\303\247\303\243o diferencial L(y) = t^4 (onde L \303\251 como acima).
A observa\303\247\303\243o importante aqui \303\251 que L(polin\303\264mio)=polin\303\264mio. (Isso fica claro olhando a defini\303\247\303\243o de L e lembrando que derivada de polin\303\264mio \303\251 polin\303\264mio). Por exemplo:
L(3*t^2+t^5);
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Outra observa\303\247\303\243o f\303\241cil \303\251 que o grau do polin\303\264mio \303\251 mantido (para o L em quest\303\243o).
Isto sugere procurar a solu\303\247\303\243o de L(y) = t^4 como um polin\303\264mio de grau 4:
L(A+B*t+C*t^2+D*t^3+E*t^4);
LEBJIkVHNiIiI0NJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQiIicqJkYjIiIiSSJ0R0YkRixGJUkiQ0dGJCEjOSomRiZGLEYtRiwhI1UqJkYjRixGLSIiIyEjJSlJIkJHRiQhIiIqJkYuRixGLUYsISIjKiZGJkYsRi1GMyEiJComRiNGLEYtIiIkISIlSSJBR0YkRioqJkY1RixGLUYsRioqJkYuRixGLUYzRioqJkYmRixGLUY8RioqJkYjRixGLSIiJUYq
collect(%,t);
LDQqJkkiRUc2IiIiIkkidEdGJSIiJSIiJyomLCZGJCEiJUkiREc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJUYpRiZGJyIiJEYmKiYsKEYkISMlKUYtISIkSSJDR0YlRilGJkYnIiIjRiYqJiwqRi0hI1VGNiEiI0YkIiNDSSJCR0YlRilGJkYnRiZGJkYkRjxGLUYpRj0hIiJGNiEjOUkiQUdGJUYp
Queremos que a sa\303\255da seja t^4. Logo temos que resolver o sistema linear:
solve({6*E=1, -4*E+6*D=0, -84*E-3*D+6*C=0, -42*D-2*C+24*E+6*B =0, 24*E+6*D-B-14*C+6*A=0});
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Est\303\241 encontrada a solu\303\247\303\243o particular. Confirmando:
L(1603/324 + 49/54*t + 43/18*t^2 + 1/9*t^3 + 1/6*t^4);
KiRJInRHNiIiIiU=