       Re: Boundary Value Problem

• To: mathgroup at smc.vnet.net
• Subject: [mg6910] Re: Boundary Value Problem
• From: Stephen P Luttrell <luttrell at signal.dra.hmg.gb>
• Date: Fri, 25 Apr 1997 14:00:40 -0400 (EDT)
• Organization: Defence Research Agency
• Sender: owner-wri-mathgroup at wolfram.com

```>
> y'''''[x]+y[x]==0
>
> Conditions
> y==0
> y'==7
>
> y==1
> y'==10
> y''=5
>

The solution to this problem that I posted earlier was flawed
(it was rubbish, to be honest!).

I can't get a solution to come out by feeding DSolve with the
differential
equation plus its boundary conditions.

However, I can obtain a solution as follows:

1. Solve the differential equation without boundary conditions:

soln=DSolve[{y'''''[x]+y[x]==0},y[x],x]

which gives:

{{y[x] -> C/E^x + E^((-1)^(1/5)*x)*C +
C/E^(-(-(-1)^(2/5)*x)) + E^((-1)^(3/5)*x)*C +
C/E^(-(-(-1)^(4/5)*x))}}

2. Define a function which implements the solution without boundary
conditions:

y[x_] := C/E^x + E^((-1)^(1/5)*x)*C +
C/E^(-(-(-1)^(2/5)*x)) + E^((-1)^(3/5)*x)*C +
C/E^(-(-(-1)^(4/5)*x))

3. Solve for the (numerical values of the) constants:

const = Chop[Solve[N[{y == 0, Derivative[y] == 7,
y == 1, Derivative[y] == 10,
Derivative[Derivative[y]] == 5}],
{C, C, C, C, C}]]

which gives:

{{C -> -191.2756607271927,
C -> 57.89644611472424 + 9.823539039817276*I,
C -> 37.74138424887295 - 127.7949066168243*I,
C -> 37.74138424887115 + 127.7949066168247*I,
C -> 57.8964461147244 - 9.82353903981644*I}}

4. Substitute the constants into the solution:

Chop[ComplexExpand[y[x] /. const[]]]

which gives:

-191.2756607271927/E^x + 115.7928922294486*E^(1/4*(1 + Sqrt)*x)*
Cos[1/2*Sqrt[1/2*(5 - Sqrt)]*x] +
75.4827684977441*E^(1/4*(1 - Sqrt)*x)*
Cos[1/2*Sqrt[1/2*(5 + Sqrt)]*x] -
19.64707807963371*E^(1/4*(1 + Sqrt)*x)*
Sin[1/2*Sqrt[1/2*(5 - Sqrt)]*x] -
255.5898132336491*E^(1/4*(1 - Sqrt)*x)*
Sin[1/2*Sqrt[1/2*(5 + Sqrt)]*x]

5. Define a function which implements the solution with boundary
conditions:

z[x_] := -191.2756607271927/E^x +
115.7928922294486*E^(1/4*(1 + Sqrt)*x)*
Cos[1/2*Sqrt[1/2*(5 - Sqrt)]*x] +
75.4827684977441*E^(1/4*(1 - Sqrt)*x)*
Cos[1/2*Sqrt[1/2*(5 + Sqrt)]*x] -
19.64707807963371*E^(1/4*(1 + Sqrt)*x)*
Sin[1/2*Sqrt[1/2*(5 - Sqrt)]*x] -
255.5898132336491*E^(1/4*(1 - Sqrt)*x)*
Sin[1/2*Sqrt[1/2*(5 + Sqrt)]*x]

6. Verify that this solution satisfies all of the required conditions:

Chop[{Derivative[Derivative[Derivative[Derivative[
Derivative[z]]]]][x] + z[x], z, Derivative[z],
z, Derivative[z], Derivative[Derivative[z]]}]

which gives:

{0, 0, 7., 1.000000000000028, 9.999999999999989, 5.}

This is pretty accurate.

==================================================================================

Dr Stephen P Luttrell                  luttrell at signal.dra.hmg.gb