# can't solve this differential equation

Posted 9 years ago
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 Hi all i have problem with solving this eq: can u please help me
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Posted 9 years ago
 Let's try the simple thing last In:= 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:= Clear[aq] aq = aliEq[Sum[a[o] x^o, {o, 1, n}], x]; In:= Coefficient[aq /. n -> 100, x, #] & /@ Range Out= {9 a a - 12 b a + 245 b a, 16 a a - 20 b a + 606 b a, 25 a a - 30 b a + 1267 b a, 36 a a - 42 b a + 2360 b a, 49 a a - 56 b a + 4041 b a, 64 a a - 72 b a + 6490 b a, 81 a a - 90 b a + 9911 b a, 100 a a - 110 b a + 14532 b a, 121 a a - 132 b a + 20605 b a, 144 a a - 156 b a + 28406 b a, 169 a a - 182 b a + 38235 b a, 196 a a - 210 b a + 50416 b a, 225 a a - 240 b a + 65297 b a, 256 a a - 272 b a + 83250 b a, 289 a a - 306 b a + 104671 b a, 324 a a - 342 b a + 129980 b a, 361 a a - 380 b a + 159621 b a, 400 a a - 420 b a + 194062 b a, 441 a a - 462 b a + 233795 b a, 484 a a - 506 b a + 279336 b a} In:= FindSequenceFunction[{245, 606, 1267, 2360, 4041, 6490, 9911, 14532, 20605, 28406, 38235, 50416}] Out= 76 + 103 #1 + 53 #1^2 + 12 #1^3 + #1^4 & In:= 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:= coeff Out= 484 a a - 506 b a + 279336 b a 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.
Posted 9 years ago
 This is not an exact equation In:= 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:= (-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= (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
Posted 9 years ago
 thank ua,b are constant.and my boundary conditions are:w[d]==0 ,w= = limit (cant be infinite i.e if there is any "r" in denominator then the Corresponding Factor must be zero for example if we have a term like this: " C /r + a*r^2 " then C 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]== limithow can i solve it with this boundary conditions?
Posted 9 years ago
 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:= 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, w' == 0, w'' == 1, w''' == 2}, w, r] then make it into a function which can be plotted In:= Clear[f] f = (Out[[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.
Posted 9 years ago
 one of the boundary condition is (D[D[w[r], r], r] + 1/r D[w[r], r]) when r=0 ,cant be infinitehow can i define it in mathematica language!?
Posted 9 years ago
 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 don't 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.
Posted 9 years ago
 DSolve doesn't give a solution. This mean there probably isn't a general symbolic to the problem that you've 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 you're trying to solve.
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Posted 9 years ago
 That's 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, w', w'', w''', 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 you're 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.