Help with rearranging the equation and solution Steps?

Questions
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1

Hi,

How can I rearrange the equation eq1 in terms of (H + Derivative[1][G][[Xi]]/G[[Xi]]^2) and then solve for it?

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2

It seems you have done it. What is wrong with it?

3

In order to obtain an algebraic system, one must set the coeffcients of (H + G′/G2)i to zero. Solve the obtained algebraic system.

But the expressions were not arranged correctly, no answer was obtained, while the answer was as follows:

4597123.PNG
4597123.PNG793×249 23.3 KB

4

Your presumed solution contain C,B,beta, while the equation has no such variables. You can check the solutions given by Mathematica:

eq1 /. Solve[
   coefficients == 0, {a0, a1, a2, a3, a4, H, \[Nu]}] // Simplify

and they are fine.

5

Yes, you are right; I made some mistakes in certain parts. I just have one question: If I want to rearrange an equation I have in terms of that expression mentioned in the first question, so that there are no functions within the equation, what should I do?

6

Sorry, I don’t understand your question.

7

I don’t know why the placement isn’t happening?

dG[\[Xi]_] := D[G[\[Xi]], \[Xi]];
\[CapitalXi]dG[\[Xi]_] := dG[\[Xi]]/G[\[Xi]]^2;
D[\[CapitalXi]dG[\[Xi]], \[Xi]] = 
  A + B \[CapitalXi]dG[\[Xi]]^2 + c \[CapitalXi]dG[\[Xi]]; 


a = {a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, 
   a15, a16, a17, a18, a19, a20};


U[\[Xi]_] := Sum[a[[i + 3]]*(H + \[CapitalXi]dG[\[Xi]])^i, {i, p, q}]
p = -1; q = 1;
U[\[Xi]]
eq1 = 2*(\[Nu]^2 - k^2)*D[U[\[Xi]], \[Xi]] + 
   2 (\[Alpha] k^2 \[Nu]^2)*D[U[\[Xi]], {\[Xi], 2}] - 
   3 \[Nu] k^2*D[U[\[Xi]], \[Xi]]^2 // Expand
8

What placement? Your third line cannot work with = . Maybe you meant it as an equation?

dG = G';
\[CapitalXi]dG[\[Xi]_] := dG[\[Xi]]/G[\[Xi]]^2;
\[CapitalXi]dG'[\[Xi]] == 
 A + B \[CapitalXi]dG[\[Xi]]^2 + c \[CapitalXi]dG[\[Xi]]
9

Yes, I want the expressions that I defined above in section eq1 to be substituted into this equation, but when I run the code, the substitution does not occur.???