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Help me to solve this equation with Solve command

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Hello, I'm trying to solve an equation as in attached image. However, Mathematica version 9 couldn't solve the equation and return with error message: "Solve::nsmet: This system cannot be solved with the methods available to Solve"

There are any other option that i missed for the Solve command? I would greatly appreciate any help you can give me in working this problem.

Solve::nsmet: This system cannot be solved with the methods available to Solve

POSTED BY: dang chau
Answer
2 months ago

(1) If you do not provide full code to replicate the example then it is impossible to really say much, short of retyping from an image (which I for one will not do).

(2) You will have a better chance if you make the upper bound an explicit number, 10, in this case, rather than encode it in the equations. Solve sees the equations but Sum does not, and Solve needs Sum to fully evaluate (to something explicit rather than a Sum).

(3) It is not a good idea to use N as a variable name, since it is also a heavily used function in Mathematica.

POSTED BY: Daniel Lichtblau
Answer
2 months ago

Thank you for your comments.

My original equation is

y = Sum_{i=0}{N} (v[i] - v0/Xs * (Xs - x[i])^3)^2

First, I get the differential for the Xs variable

y1 = D[y,Xs]

Then, I'd like to solve the above equation for Xs by using Solve command:

Solve[y1,Xs,Reals]

I also try to solve the original equation by hand, however i still got the final result because it is to complicated long equation.

I would greatly appreciate any help you can give me in working this problem.

POSTED BY: dang chau
Answer
2 months ago

Perhaps this

n = 10; v0 = 20;
y = Sum[(v[i] - v0/Xs*(Xs - x[i])^3)^2, {i, 0, n}];
y1 = Expand[D[y, Xs] Xs^3];(*Remove Xs from denominators*)
Simplify[Reduce[y1 == 0, Xs]]

All of the details of determining the sixth power polynomial in Xs has been done for you. With your values for x[i] and v[j] perhaps you can determine which roots are Real.

Or if the Root notation is confusing then use your values of x[i] and v[j] to determine the real roots of

440 Xs^6 -
Xs^5 Sum[180 x[i], {i, 0, 10}] -
Xs^4 Sum[2 v[i] - 300 x[i]^2, {i, 0, 10}] +
Xs^3 Sum[3 x[i] v[i] - 200 x[i]^3, {i, 0, 10}] -
Xs Sum[x[i]^3 v[i] - 60 x[i]^5, {i, 0, 10}] -
Sum[20 x[i]^6, {i, 0, 10}]

which is just a manual reformatting of the result from the code above.

Plot might even help understanding how many real roots there are or Reduce might do that if you define all your x[i] and v[j] values first.

POSTED BY: Bill Simpson
Answer
2 months ago