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]==0 > y'[0]==7 > > y[1]==1 > y'[1]==10 > y''[1]=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[1]/E^x + E^((-1)^(1/5)*x)*C[2] + C[3]/E^(-(-(-1)^(2/5)*x)) + E^((-1)^(3/5)*x)*C[4] + C[5]/E^(-(-(-1)^(4/5)*x))}} 2. Define a function which implements the solution without boundary conditions: y[x_] := C[1]/E^x + E^((-1)^(1/5)*x)*C[2] + C[3]/E^(-(-(-1)^(2/5)*x)) + E^((-1)^(3/5)*x)*C[4] + C[5]/E^(-(-(-1)^(4/5)*x)) 3. Solve for the (numerical values of the) constants: const = Chop[Solve[N[{y[0] == 0, Derivative[1][y][0] == 7, y[1] == 1, Derivative[1][y][1] == 10, Derivative[1][Derivative[1][y]][1] == 5}], {C[1], C[2], C[3], C[4], C[5]}]] which gives: {{C[1] -> -191.2756607271927, C[2] -> 57.89644611472424 + 9.823539039817276*I, C[3] -> 37.74138424887295 - 127.7949066168243*I, C[4] -> 37.74138424887115 + 127.7949066168247*I, C[5] -> 57.8964461147244 - 9.82353903981644*I}} 4. Substitute the constants into the solution: Chop[ComplexExpand[y[x] /. const[[1]]]] which gives: -191.2756607271927/E^x + 115.7928922294486*E^(1/4*(1 + Sqrt[5])*x)* Cos[1/2*Sqrt[1/2*(5 - Sqrt[5])]*x] + 75.4827684977441*E^(1/4*(1 - Sqrt[5])*x)* Cos[1/2*Sqrt[1/2*(5 + Sqrt[5])]*x] - 19.64707807963371*E^(1/4*(1 + Sqrt[5])*x)* Sin[1/2*Sqrt[1/2*(5 - Sqrt[5])]*x] - 255.5898132336491*E^(1/4*(1 - Sqrt[5])*x)* Sin[1/2*Sqrt[1/2*(5 + Sqrt[5])]*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[5])*x)* Cos[1/2*Sqrt[1/2*(5 - Sqrt[5])]*x] + 75.4827684977441*E^(1/4*(1 - Sqrt[5])*x)* Cos[1/2*Sqrt[1/2*(5 + Sqrt[5])]*x] - 19.64707807963371*E^(1/4*(1 + Sqrt[5])*x)* Sin[1/2*Sqrt[1/2*(5 - Sqrt[5])]*x] - 255.5898132336491*E^(1/4*(1 - Sqrt[5])*x)* Sin[1/2*Sqrt[1/2*(5 + Sqrt[5])]*x] 6. Verify that this solution satisfies all of the required conditions: Chop[{Derivative[1][Derivative[1][Derivative[1][Derivative[1][ Derivative[1][z]]]]][x] + z[x], z[0], Derivative[1][z][0], z[1], Derivative[1][z][1], Derivative[1][Derivative[1][z]][1]}] which gives: {0, 0, 7., 1.000000000000028, 9.999999999999989, 5.} This is pretty accurate. ================================================================================== Dr Stephen P Luttrell luttrell at signal.dra.hmg.gb Adaptive Systems Theory 01684-894046 (phone) Room EX21, Defence Research Agency 01684-894384 (fax) Malvern, Worcs, WR14 3PS, U.K. http://www.dra.hmg.gb/cis5pip/Welcome.html