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Question: How could I substitute the (t1*e) into t2 at the index of e? I am trying many ways to do that, but it doesn't work. please help me.

Question:How could I substitute the (t1*e) into t2 at the index of e? I am trying many ways to do that, but it doesn't work. please help me.

ssllys 35 Maple 2018
substitution subs
July 27 2018
0

How could I substitute the (t1*e) into t2 at the index of e?  I am trying many ways to do that, but it doesn't work. please help me.
 

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restart:
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alias(epsilon = e, omega = w, omega[0] = w0, t[1] = t1, t[2] = t2, alpha[1] = a1, alpha[2] = a2, alpha[3] = a3, beta[1] = b1, beta[2] = b2, F[0] = f0); e := proc (t1, t2) options operator, arrow; e end proc; w0 := proc (t1, t2) options operator, arrow; w0 end proc; a := proc (t1, t2) options operator, arrow; a end proc; f := proc (t1, t2) options operator, arrow; f end proc; mu := proc (t1, t2) options operator, arrow; mu end proc
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(1/8)*alpha[2](t[1], t[2])*b^3*exp((3*I)*omega*t[1])+(1/8)*alpha[2](t[1], t[2])*b^3*exp(-(3*I)*omega*t[1])+(1/8)*alpha[2](t[1], t[2])*A(t[2])^3*exp((3*I)*t[1])+(1/8)*alpha[2](t[1], t[2])*A(t[2])^3*exp(-(3*I)*t[1])+3*alpha[2](t[1], t[2])*b^3*exp(-I*omega*t[1])*(1/8)+3*alpha[2](t[1], t[2])*b^3*exp(I*omega*t[1])*(1/8)-I*alpha[1](t[1], t[2])*A(t[2])*exp(-I*t[1])*(1/2)+3*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(-I*omega*t[1])*(1/4)-I*(diff(A(t[2]), t[2]))*exp(-I*t[1])+3*alpha[2](t[1], t[2])*A(t[2])*exp(-I*t[1])*b^2*(1/4)-I*alpha[1](t[1], t[2])*b*omega*exp(-I*omega*t[1])*(1/2)+I*alpha[1](t[1], t[2])*b*omega*exp(I*omega*t[1])*(1/2)+3*alpha[2](t[1], t[2])*A(t[2])^3*exp(-I*t[1])*(1/8)+I*(diff(A(t[2]), t[2]))*exp(I*t[1])+3*alpha[2](t[1], t[2])*A(t[2])*b^2*exp(-I*t[1]-(2*I)*omega*t[1])*(1/8)+3*alpha[2](t[1], t[2])*A(t[2])*b^2*exp(-I*t[1]+(2*I)*omega*t[1])*(1/8)+q[1](t[1], t[2])+3*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(-(2*I)*t[1]-I*omega*t[1])*(1/8)+3*alpha[2](t[1], t[2])*A(t[2])^3*exp(I*t[1])*(1/8)+3*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(-(2*I)*t[1]+I*omega*t[1])*(1/8)+3*alpha[2](t[1], t[2])*A(t[2])*b^2*exp(I*t[1]-(2*I)*omega*t[1])*(1/8)+3*alpha[2](t[1], t[2])*A(t[2])*b^2*exp(I*t[1]+(2*I)*omega*t[1])*(1/8)+diff(q[1](t[1], t[2]), t[1], t[1])+3*alpha[2](t[1], t[2])*A(t[2])^2*b*exp((2*I)*t[1]-I*omega*t[1])*(1/8)+3*alpha[2](t[1], t[2])*A(t[2])^2*b*exp((2*I)*t[1]+I*omega*t[1])*(1/8)+I*alpha[1](t[1], t[2])*A(t[2])*exp(I*t[1])*(1/2)+3*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(I*omega*t[1])*(1/4)+3*alpha[2](t[1], t[2])*A(t[2])*exp(I*t[1])*b^2*(1/4);

(3/4)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(-I*omega*t[1])+(3/4)*alpha[2](t[1], t[2])*A(t[2])*exp(-I*t[1])*b^2+(3/8)*alpha[2](t[1], t[2])*A(t[2])*b^2*exp(-I*t[1]-(2*I)*omega*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])*b^2*exp(-I*t[1]+(2*I)*omega*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(-(2*I)*t[1]-I*omega*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(-(2*I)*t[1]+I*omega*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])*b^2*exp(I*t[1]-(2*I)*omega*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])*b^2*exp(I*t[1]+(2*I)*omega*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp((2*I)*t[1]-I*omega*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp((2*I)*t[1]+I*omega*t[1])+(3/4)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(I*omega*t[1])+(3/4)*alpha[2](t[1], t[2])*A(t[2])*exp(I*t[1])*b^2-((1/2)*I)*alpha[1](t[1], t[2])*A(t[2])*exp(-I*t[1])+((1/2)*I)*alpha[1](t[1], t[2])*A(t[2])*exp(I*t[1])+I*(diff(A(t[2]), t[2]))*exp(I*t[1])+(1/8)*alpha[2](t[1], t[2])*b^3*exp((3*I)*omega*t[1])+(1/8)*alpha[2](t[1], t[2])*b^3*exp(-(3*I)*omega*t[1])+(1/8)*alpha[2](t[1], t[2])*A(t[2])^3*exp((3*I)*t[1])+(1/8)*alpha[2](t[1], t[2])*A(t[2])^3*exp(-(3*I)*t[1])+(3/8)*alpha[2](t[1], t[2])*b^3*exp(-I*omega*t[1])+(3/8)*alpha[2](t[1], t[2])*b^3*exp(I*omega*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])^3*exp(-I*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])^3*exp(I*t[1])-I*(diff(A(t[2]), t[2]))*exp(-I*t[1])+((1/2)*I)*alpha[1](t[1], t[2])*b*omega*exp(I*omega*t[1])-((1/2)*I)*alpha[1](t[1], t[2])*b*omega*exp(-I*omega*t[1])+diff(diff(q[1](t[1], t[2]), t[1]), t[1])+q[1](t[1], t[2])

 (1)
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> 

 

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eq:=combine(subs(w=3+e*sigma,%),exp);

(3/4)*alpha[2](t[1], t[2])*A(t[2])*exp(-I*t[1])*b^2+(3/4)*alpha[2](t[1], t[2])*A(t[2])*exp(I*t[1])*b^2-((1/2)*I)*alpha[1](t[1], t[2])*A(t[2])*exp(-I*t[1])+((1/2)*I)*alpha[1](t[1], t[2])*A(t[2])*exp(I*t[1])+(3/4)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(-I*(epsilon*sigma+3)*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])*b^2*exp(-I*t[1]-(2*I)*(epsilon*sigma+3)*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])*b^2*exp(-I*t[1]+(2*I)*(epsilon*sigma+3)*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(-(2*I)*t[1]-I*(epsilon*sigma+3)*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(-(2*I)*t[1]+I*(epsilon*sigma+3)*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])*b^2*exp(I*t[1]-(2*I)*(epsilon*sigma+3)*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])*b^2*exp(I*t[1]+(2*I)*(epsilon*sigma+3)*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp((2*I)*t[1]-I*(epsilon*sigma+3)*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp((2*I)*t[1]+I*(epsilon*sigma+3)*t[1])+(3/4)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(I*(epsilon*sigma+3)*t[1])+I*(diff(A(t[2]), t[2]))*exp(I*t[1])+(1/8)*alpha[2](t[1], t[2])*A(t[2])^3*exp((3*I)*t[1])+(1/8)*alpha[2](t[1], t[2])*A(t[2])^3*exp(-(3*I)*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])^3*exp(-I*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])^3*exp(I*t[1])-I*(diff(A(t[2]), t[2]))*exp(-I*t[1])+(1/8)*alpha[2](t[1], t[2])*b^3*exp((3*I)*(epsilon*sigma+3)*t[1])+(3/8)*alpha[2](t[1], t[2])*b^3*exp(I*(epsilon*sigma+3)*t[1])+(1/8)*alpha[2](t[1], t[2])*b^3*exp(-(3*I)*(epsilon*sigma+3)*t[1])+(3/8)*alpha[2](t[1], t[2])*b^3*exp(-I*(epsilon*sigma+3)*t[1])+((1/2)*I)*alpha[1](t[1], t[2])*b*(epsilon*sigma+3)*exp(I*(epsilon*sigma+3)*t[1])-((1/2)*I)*alpha[1](t[1], t[2])*b*(epsilon*sigma+3)*exp(-I*(epsilon*sigma+3)*t[1])+diff(diff(q[1](t[1], t[2]), t[1]), t[1])+q[1](t[1], t[2])

 (2)
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expand(%);

(3/4)*alpha[2](t[1], t[2])*A(t[2])*exp(I*t[1])*b^2+((1/2)*I)*alpha[1](t[1], t[2])*A(t[2])*exp(I*t[1])+I*(diff(A(t[2]), t[2]))*exp(I*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])^3*exp(I*t[1])+(1/8)*alpha[2](t[1], t[2])*A(t[2])^3*(exp(I*t[1]))^3+(1/8)*alpha[2](t[1], t[2])*A(t[2])^3/(exp(I*t[1]))^3+(3/8)*alpha[2](t[1], t[2])*A(t[2])^3/exp(I*t[1])-I*(diff(A(t[2]), t[2]))/exp(I*t[1])+diff(diff(q[1](t[1], t[2]), t[1]), t[1])+q[1](t[1], t[2])+(1/8)*alpha[2](t[1], t[2])*b^3*(exp(I*epsilon*sigma*t[1]))^3*(exp(I*t[1]))^9+(3/8)*alpha[2](t[1], t[2])*b^3*exp(I*epsilon*sigma*t[1])*(exp(I*t[1]))^3+(1/8)*alpha[2](t[1], t[2])*b^3/((exp(I*epsilon*sigma*t[1]))^3*(exp(I*t[1]))^9)+(3/8)*alpha[2](t[1], t[2])*b^3/(exp(I*epsilon*sigma*t[1])*(exp(I*t[1]))^3)+(3/4)*alpha[2](t[1], t[2])*A(t[2])*b^2/exp(I*t[1])-((1/2)*I)*alpha[1](t[1], t[2])*A(t[2])/exp(I*t[1])+(3/4)*alpha[2](t[1], t[2])*A(t[2])^2*b/(exp(I*epsilon*sigma*t[1])*(exp(I*t[1]))^3)+(3/8)*alpha[2](t[1], t[2])*A(t[2])*b^2/((exp(I*epsilon*sigma*t[1]))^2*(exp(I*t[1]))^7)+(3/8)*alpha[2](t[1], t[2])*A(t[2])*b^2*(exp(I*epsilon*sigma*t[1]))^2*(exp(I*t[1]))^5+(3/8)*alpha[2](t[1], t[2])*A(t[2])^2*b/(exp(I*epsilon*sigma*t[1])*(exp(I*t[1]))^5)+(3/8)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(I*epsilon*sigma*t[1])*exp(I*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])*b^2/((exp(I*epsilon*sigma*t[1]))^2*(exp(I*t[1]))^5)+(3/8)*alpha[2](t[1], t[2])*A(t[2])*b^2*(exp(I*epsilon*sigma*t[1]))^2*(exp(I*t[1]))^7+(3/8)*alpha[2](t[1], t[2])*A(t[2])^2*b/(exp(I*epsilon*sigma*t[1])*exp(I*t[1]))+(3/8)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(I*epsilon*sigma*t[1])*(exp(I*t[1]))^5+(3/4)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(I*epsilon*sigma*t[1])*(exp(I*t[1]))^3+((3/2)*I)*alpha[1](t[1], t[2])*b*exp(I*epsilon*sigma*t[1])*(exp(I*t[1]))^3-((3/2)*I)*alpha[1](t[1], t[2])*b/(exp(I*epsilon*sigma*t[1])*(exp(I*t[1]))^3)+((1/2)*I)*alpha[1](t[1], t[2])*b*exp(I*epsilon*sigma*t[1])*(exp(I*t[1]))^3*epsilon*sigma-((1/2)*I)*alpha[1](t[1], t[2])*b*epsilon*sigma/(exp(I*epsilon*sigma*t[1])*(exp(I*t[1]))^3)

 (3)
> 

eq:=combine(subs({e^(t1*e)=e^t2},%),exp);

(3/4)*alpha[2](t[1], t[2])*A(t[2])*exp(-I*t[1])*b^2+(3/4)*alpha[2](t[1], t[2])*A(t[2])*exp(I*t[1])*b^2-((1/2)*I)*alpha[1](t[1], t[2])*A(t[2])*exp(-I*t[1])+((1/2)*I)*alpha[1](t[1], t[2])*A(t[2])*exp(I*t[1])+(1/8)*alpha[2](t[1], t[2])*b^3*exp((3*I)*t[1]*epsilon*sigma+(9*I)*t[1])+(3/8)*alpha[2](t[1], t[2])*b^3*exp(I*epsilon*sigma*t[1]+(3*I)*t[1])+(1/8)*alpha[2](t[1], t[2])*b^3*exp(-(3*I)*t[1]*epsilon*sigma-(9*I)*t[1])+(3/8)*alpha[2](t[1], t[2])*b^3*exp(-I*epsilon*sigma*t[1]-(3*I)*t[1])+I*(diff(A(t[2]), t[2]))*exp(I*t[1])+(1/8)*alpha[2](t[1], t[2])*A(t[2])^3*exp((3*I)*t[1])+(1/8)*alpha[2](t[1], t[2])*A(t[2])^3*exp(-(3*I)*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])^3*exp(-I*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])^3*exp(I*t[1])-I*(diff(A(t[2]), t[2]))*exp(-I*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])*b^2*exp(-(2*I)*epsilon*sigma*t[1]-(7*I)*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])*b^2*exp((2*I)*epsilon*sigma*t[1]+(5*I)*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(-I*epsilon*sigma*t[1]-(5*I)*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(I*epsilon*sigma*t[1]+I*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])*b^2*exp(-(2*I)*epsilon*sigma*t[1]-(5*I)*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])*b^2*exp((2*I)*epsilon*sigma*t[1]+(7*I)*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(-I*epsilon*sigma*t[1]-I*t[1])+(3/8)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(I*epsilon*sigma*t[1]+(5*I)*t[1])+diff(diff(q[1](t[1], t[2]), t[1]), t[1])+q[1](t[1], t[2])+(3/4)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(-I*epsilon*sigma*t[1]-(3*I)*t[1])+(3/4)*alpha[2](t[1], t[2])*A(t[2])^2*b*exp(I*epsilon*sigma*t[1]+(3*I)*t[1])+((3/2)*I)*alpha[1](t[1], t[2])*b*exp(I*epsilon*sigma*t[1]+(3*I)*t[1])-((3/2)*I)*alpha[1](t[1], t[2])*b*exp(-I*epsilon*sigma*t[1]-(3*I)*t[1])+((1/2)*I)*alpha[1](t[1], t[2])*b*epsilon*sigma*exp(I*epsilon*sigma*t[1]+(3*I)*t[1])-((1/2)*I)*alpha[1](t[1], t[2])*b*epsilon*sigma*exp(-I*epsilon*sigma*t[1]-(3*I)*t[1])

 (4)
> 

How could I substitute the (t1*e) into t2 at the index of e?  I am trying many ways to do that, but it doesn't work. please help me.   (t1*e)=t2


 

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