Exp-Trig Manipulation
- To: mathgroup at smc.vnet.net
- Subject: [mg59358] Exp-Trig Manipulation
- From: Daniele Lupo <danwolf80_no_spam_please_ at libero.it>
- Date: Sun, 7 Aug 2005 03:46:53 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Hi to everyone.
I'd like to know how I can convert this expression
E^((2 - I)*x)*C[1] + E^((2 + I)*x)*C[2] + E^((3 - 4*I)*x)*C[3] + E^((3 +
4*I)*x)*C[4]
in its equivalent form
E^(2*x)*C[2]*Cos[x] + E^(3*x)*C[4]*Cos[4*x] + E^(2*x)*C[1]*Sin[x] +
E^(3*x)*C[3]*Sin[4*x]
I've obtained them while resolving a differential equation. I've tried to
solve this:
car = y''''[x] - 10*y'''[x] + 54*y''[x] - 130*y'[x] + 125*y[x] == 0;
In two different ways: first, working with characteristic polynomial:
-------------------
(* Conversion from differential equation to characteristic polynomial *)
pol = car /. {Derivative[n_][y][x] -> ë^n, y[x] -> 1};
(* Solutiof of c.p. *)
sol = Solve[pol, ë];
(* Mapping solutions in a linear combination of exponentials *)
solution1 = Plus @@ MapIndexed[C[#2[[1]]]*Exp[x*#1] & , ë /. %]
-------------------
While I've obtained second solution using DSolve:
-------------------
solution2 = y[x] /. DSolve[car, y[x], x][[1]]
-------------------
So, if I did not wrong something, these two solutions must be equivalent,
but I can't find a way to trasform solution1 to solution2: I know that
there can be a problem in conversion of C[n] coefficients during
transformation, but I don't care it. I'd like instead to convert in the
right way exponentials of first method in right product of Cos, Sin, Exp of
the solution obtained with DSolve.
Thanks for answers
Daniele
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