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I'm trying to solve the following differential equation with some specific conditions

NDSolve[{2 V[X] - 2 X Derivative[1][V][X] - 
X^2 V[X] Derivative[1][V][X] + 4 X^3 V[X] Derivative[1][V][X] - 
X^3 Derivative[1][V][X]^2 + 2 X^4 Derivative[1][V][X]^2 - 
X^3 V[X] (V^\[Prime]\[Prime])[X] + 
2 X^4 V[X] (V^\[Prime]\[Prime])[X] == 0, V[0] == 0, 
V'[0] == 0}, V, {X, 0, 10^3}]

But I have error messages :

Power: Infinite expression 1/0. encountered.

Power: Infinite expression 1/0.^3 encountered.

Infinity::indet: Indeterminate expression 0. ComplexInfinity ComplexInfinity encountered.

NDSolve::ndnum: Encountered non-numerical value for a derivative at X == 0.`.

How could I go around these problems and solve them ?

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  • 1
    $\begingroup$ One technique is, instead of starting the integration at the singularity at X == 0, to start near it, say at X = $MachineEpsilon. $\endgroup$ – Michael E2 Nov 5 '18 at 16:04
  • $\begingroup$ In your case you'll need a starting value for X and V[X] so that Abs[X^4 V[X]] is somewhat larger than $MachineEpsilon....There are probably other ways to go at it, too. $\endgroup$ – Michael E2 Nov 5 '18 at 20:12
  • $\begingroup$ The only solution with boundary conditions V[0] == 0, V'[0] == 0 is V[X] = 0. $\endgroup$ – Alex Trounev Nov 5 '18 at 23:59

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