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Here I have one problem which should help me to understand how to transform the system of differential equations with the condition to neglect two of four solutions and to get the appropriate system of equations with reduced order of differential equations but with the exactly the same two solutions with two chosen solutions of source system.


Let's see what was my idea and how I solved homogenous problem. The source system has the following form

system 1)

first eq

  a1*X1[x] + a2*X2[x] + a3*Derivative[2][X1][x] - 
  a4*Derivative[2][X2][x] + a5*Derivative[4][X1][x]=0;

second eq

  b2*X2[x] + b1*X1[x] - b4*Derivative[2][X2][x] + 
  b3*Derivative[2][X1][x] + b5*Derivative[4][X2][x]=0;

This system has 4 +/- solutions (8 general). Solutions are in the form X1[x]=C*e^(j*v*x), X2[x]=D*e^(j*v*x) where is (j^2=-1). So firstly I got analytically solutions without initial conditions, just v. Then I choose 2 positive which I need, another two positive are not important and rest 4 negative also. Then we have known solutions which I need v1=f[a1,a2...] and v2=f[a1,a2,...]. These solutions should be now solutions of system of two equations second order (not 4 anymore) and I made system of two new equations second order to have exactly the same two solutions, of course another two because of the nature of the system are negative and not important but the same. So my new system is in the form with new constants now in the form

system 2) appropriate

first new eq

  A1*X1[x] + A2*X2[x] + A3*Derivative[2][X1][x]=0;

second new eq

   B1*X2[x] + B2*X1[x] + B3*Derivative[2][X2][x]=0;

My condition was to have this form of diff equations above, and solutions of this new system are 4, (2 positive, 2 negative the same). These two solutions are also solutions of the started system. I solved problem for homogenous system.

My question now is how to transform non-homogenous system to have appropriate reduced system with the same conditions as above described (2 solutions the same v1 and v2, initial conditions are not important, just v)

Now I have system

system 3) first eq

  g1[x]==a1*X1[x] + a2*X2[x] + a3*Derivative[2][X1][x] - 
  a4*Derivative[2][X2][x] + a5*Derivative[4][X1][x];

second eq

  g2[x]==b2*X2[x] + b1*X1[x] - b4*Derivative[2][X2][x] + 
  b3*Derivative[2][X1][x] + b5*Derivative[4][X2][x];

How to get new non-homogenous system in the following form

first new eq

   A1*X1[x] + A2*X2[x] + A3*Derivative[2][X1][x]=Q1*g1[x];

second new eq

   B1*X2[x] + B2*X1[x] + B3*Derivative[2][X2][x]=Q2*g2[x];

Q1 and Q2 are the problem how to find to satisfy above conditions. A1, A2, A3, B1, B2 and B3 I already found in homogenous system and I need Q1 and Q2?

Or if somebody has another idea to get new system in the form with condition to have two of four solutions the same as in case of system 3)

first new eq

   AA1*X1[x] + AA2*X2[x] + AA3*Derivative[2][X1][x]=QQ1*g1[x];

second new eq

   BB1*X2[x] + BB2*X1[x] + BB3*Derivative[2][X2][x]=QQ2*g2[x];

AA1, AA2, AA3, QQ1, BB1, BB2, BB3 and QQ2 I am looking for.

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