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# can't solve this differential equation

Posted 10 years ago
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Posted 10 years ago
 Let's 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.
Posted 10 years ago
 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
Posted 10 years ago
 thank ua,b are constant.and my boundary conditions are:w[d]==0 ,w[0]= = 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[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]== limithow can i solve it with this boundary conditions?
Posted 10 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[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.
Posted 10 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 10 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 SchrÃ¶dinger to solve the SchrÃ¶dinger equation for the hydrogen atom.
Posted 10 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.
Posted 10 years ago
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Posted 10 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.