Hi all
i have problem with solving this eq:
can u please help me
DSolve doesnât give a solution. This mean there probably isnât a general symbolic to the problem that youâve specified.
You may want to try:
- Using Assumptions to give more information about the problem.
- Numerically Solving the differential equation
What you do of course very much on the problem youâre trying to solve.
Thatâs an ordinary differential equation of fourth degree for w[r] with inhomogenity p[r]. To make this work you need to specify
- an existence interval for the independent variable r
- four boundary conditions for w, wâ, wââ, wâââ, see OrdinaryDifferentialEquation
- the inhomogenity
p[r]as a function - a and b
Then most probably you have to use NDSolve because of p[r].
If youâre a lover of step-wise thinking try to solve the homogenous equation (p[r] === 0) first.
Be aware of the singularity if r = 0 belongs to the existence interval.
one of the boundary condition is
(D[D[w[r], r], r] + 1/r D[w[r], r]) when r=0 ,cant be infinite
how can i define it in mathematica language!?
one of the boundary condition is (D[D[w[r], r], r] + 1/r D[w[r], r])
when r=0 ,cant be infinite
Still we donât know what the existence interval for the independent variable r is. If we know it, recognizing such a condition (finiteness condition, being zero at inifinity etc.) is a matter of a so called ansatz (a.k.a. educated guess).
Possibly you know the most famous example: Hermann Weyl helped Erwin Schrödinger to solve the Schrödinger equation for the hydrogen atom.
The homogeneous equation can be solved to give a DifferentialRoot object, to plot something one has to prescribe the values mentioned - done by chance here -
In[13]:= Clear[a, b, r]
DSolve[{(b + a r^2) w'[r] + (a r - b) r^2 w''[r] + 2 b r^2 w'''[r] + b r^3 w''''[r] == 0,
w[1] == 1, w'[1] == 0, w''[1] == 1, w'''[1] == 2}, w, r]
then make it into a function which can be plotted
In[50]:= Clear[f]
f = (Out[14][[1, 1, 2]]) /. {a -> 2, b -> 3}
and plot it
the behaviour depends strongly on a and b. Experiment on your own.
Another thing to do (a.k.a. jobs for the weekend): Specify a and b in an attempt to reach the form of some known (named) differential equation. Check the results. Construct solutions from known solutions.
thank u
a,b are constant.
and my boundary conditions are:
w[d]==0
,
w[0]= = limit (cant be infinite i.e if there is any ârâ in denominator then the Corresponding Factor must be zero
for example if we have a term like this: " C[1] /r + a*r^2 " then C[1] must be zero for the sake of not being infinite
,
D[D[w[r], r], r] + 1/r D[w[r], r]== limit
,
D[b (D[D[w[r], r], r] + 1/r D[w[r], r]), r]== limit
how can i solve it with this boundary conditions?
This is not an exact equation
In[160]:= Clear[a0, a1, a2, a3, a4]
a0[r_] := b
a1[r_] := 2 b/r
a2[r_] := (a - b/r)
a3[r_] := (b + a r^2)/r^3
a4[r_] := 0
In[167]:= (-1)^4 D[a0[r], {r, 4}] + (-1)^3 D[a1[r], {r, 3}] + (-1)^2 D[a2[r], {r, 2}] + (-1)^1 D[a3[r], {r, 1}] + a4[r] // Simplify
Out[167]= (15 b - 2 b r + a r^2)/r^4
unless a and b are both zero. So one cannot reduce the order of the equation. Nevertheless - in experimental mode - set v[r] = r^3 w'[r] to arrive at
(* Ansatz: w'[r] = r^3 v[r] *)
(b + a r^2) v[r] + (a - b/r) D[r^3 v[r], r] +
2 b/r D[r^3 v[r], {r, 2}] + b D[r^3 v[r], {r, 3}] // Simplify
(b (19 - 3 r) + 4 a r^2) v[r] + r ((30 b - b r + a r^2) v'[r] + b r (11 (v''[r] + r v'''[r]))
for this new equation it is possible to get some inhomogenous solutions (for simple inhomogenities) but it seems impossible to find one which is finite at r = 0.
(* inhomogen, transformiert *)
Clear[s, a, b, v, w, r]
s = With[{d = 2, \[Mu] = 1, \[Nu] = 1},
DSolve[{(b (19 - 3 r) + 4 a r^2) v[r] +
r (30 b - b r + a r^2) v'[r] + 11 b r^2 v''[r] +
b r^3 v'''[r] == r^\[Mu] (d - r)^\[Nu], v[d] == 0, v'[d] == 2,
v''[d] == 1}, v, r]]
Clear[f]
f = s[[1, 1, 2]] /. {a -> 1, b -> 2}
Plot[f[x], {x, -1, 5}]
Things which did not work
- prescribing a finiteness condition at r = 0
- get solutions to the inhomogenous equation in less than an hour (computation aborted) with NDSolve
- get solutions to the original equation for simple inhomogenities with DSolve
Letâs try the simple thing last
In[208]:= Clear[aliEq]
aliEq[f_, x_] := (b + a x^2)/x^3 D[f, x] + (a - b/x) D[f, {x, 2}] + 2 b/x D[f, {x, 3}] + b D[f, {x, 4}] /; ! FreeQ[f, x]
In[210]:= Clear[aq]
aq = aliEq[Sum[a[o] x^o, {o, 1, n}], x];
In[212]:= Coefficient[aq /. n -> 100, x, #] & /@ Range[20]
Out[212]= {9 a a[3] - 12 b a[4] + 245 b a[5],
16 a a[4] - 20 b a[5] + 606 b a[6],
25 a a[5] - 30 b a[6] + 1267 b a[7],
36 a a[6] - 42 b a[7] + 2360 b a[8],
49 a a[7] - 56 b a[8] + 4041 b a[9],
64 a a[8] - 72 b a[9] + 6490 b a[10],
81 a a[9] - 90 b a[10] + 9911 b a[11],
100 a a[10] - 110 b a[11] + 14532 b a[12],
121 a a[11] - 132 b a[12] + 20605 b a[13],
144 a a[12] - 156 b a[13] + 28406 b a[14],
169 a a[13] - 182 b a[14] + 38235 b a[15],
196 a a[14] - 210 b a[15] + 50416 b a[16],
225 a a[15] - 240 b a[16] + 65297 b a[17],
256 a a[16] - 272 b a[17] + 83250 b a[18],
289 a a[17] - 306 b a[18] + 104671 b a[19],
324 a a[18] - 342 b a[19] + 129980 b a[20],
361 a a[19] - 380 b a[20] + 159621 b a[21],
400 a a[20] - 420 b a[21] + 194062 b a[22],
441 a a[21] - 462 b a[22] + 233795 b a[23],
484 a a[22] - 506 b a[23] + 279336 b a[24]}
In[217]:= FindSequenceFunction[{245, 606, 1267, 2360, 4041, 6490, 9911, 14532, 20605, 28406, 38235, 50416}]
Out[217]= 76 + 103 #1 + 53 #1^2 + 12 #1^3 + #1^4 &
In[228]:= Clear[coeff]
coeff[n_] := (n + 2)^2 a a[n + 2] - (n + 2) (n + 3) b a[n + 3] + (76 + 103 n + 53 n^2 + 12 n^3 + n^4) b a[n + 4];
In[230]:= coeff[20]
Out[230]= 484 a a[22] - 506 b a[23] + 279336 b a[24]
the job is to solve all this equations to 0 in the homogenous case or to be equal to the corresponding coefficient of the inhomogenity p[r]. The system seems not to decouple, so the differential equation remains unsolved.