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[MUG] Solving ODE using method of undermined coefficients
| [MUG] Solving ODE using method of undermined coefficients |
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Author: Chuck Baker
Posted: 09/12/2000 05:41:57 GMT
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>> From: Chuck Baker
I've been trying to solve the following via method of undetermined coefficients:
y'' + 5y' + 6y = 3e^(-2x) + e^(3x)
The answer is the following using standard dsolve:
> deq:=diff(y(x),x,x)+5*diff(y(x),x)+6*y(x)=3*exp(-2*x)+exp(3*x):
> dsolve(deq,y(x));
y(x) = 3 exp(-2 x) x + 1/30 exp(3 x) - 3 exp(-2 x) + _C1 exp(-2 x)
+ _C2 exp(-3 x)
> odetest(%,deq); #test of the solution
0
Now, I want to be able to solve by undetermined coefficients.
First, I solve the corresponding homogeneous equation
hom_sol := dsolve(diff(y(x),`$`(x,2))+5*diff(y(x),x)+6*y(x) = 0)
y(x) = _C1*exp(-2*x)+_C2*exp(-3*x)
Then, find the yp coefficient to the equation.
A := 'A':
yp := proc (x) options operator, arrow; A*x*exp(-2*x)+x*exp(-3*x) end proc:
Now substituting yp into the equation:
eqn := simplify(diff(yp(x),`$`(x,2))+5*diff(yp(x),x)+6*yp(x) =
3*exp(-2*x)+exp(3*x))
eqn := A*exp(-2*x)-exp(-3*x) = 3*exp(-2*x)+exp(3*x)
I'm not getting an answer that leads to the same answer that is derived when
doing
a standard dsolve.
Thank very much,
Chuck
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| [MUG] Re: Solving ODE using method of undermined coefficients |
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Author: Maple Group
Posted: 15/12/2000 19:52:11 GMT
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>> From: Maple Group
| >> From: Chuck Baker
| I've been trying to solve the following via method of
| undetermined coefficients:
|
| y'' + 5y' + 6y = 3e^(-2x) + e^(3x)
|
-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-
Date: Mon, 11 Dec 2000 11:07:49 -0800 (PST)
From: Robert Israel
To:
Subject: Solving ODE using method of undermined coefficients
dsolve is right. You're using a wrong trial solution. Try
yp(x) = A*x*exp(-2*x)+B*exp(3*x).
Robert Israel
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-
Date: Tue, 12 Dec 2000 09:03:02 +0100
From: Heike Koch-Beuttenmueller
To:
Subject: Solving ODE using method of undermined coefficients
I would have done it like this:
> yp := proc (x) options operator, arrow; A*x*exp(-2*x)+B*exp(3*x) end
proc;
yp := x -> A x exp(-2 x) + B exp(3 x)
> diff(yp(x),`$`(x,2))+5*diff(yp(x),x)+6*yp(x) =
> 3*exp(-2*x)+exp(3*x);
>
A exp(-2 x) + 30 B exp(3 x) = 3 exp(-2 x) + exp(3 x)
=>
B=1/30, A=3
Heike
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