I would like to solve the following differential equation with NDSolve:

NDSolve[{(1 + Sqrt[\[Pi] x])*y[x] + x*y'[x] - (x^2)/6*y''[x] == 1, y[0] == 1, y'[0] == 0}, y[x], {x, 0, 2}]

Mathematica says it encounters an infinite expression. I've tried shifting the initial point from 0 to 0.1 or 0.01 and it gives a growing exponential function that I think is nonphysical. This DE is an approximation to another DE that has as a solution MittagLefflerE[q,-x^q] with q = 3/2 here, which is decaying. The approximation should hold for 0 < x < 2.

I also tried coding up the solution using an explicit Euler scheme but it gives the same exponential-type solution. I'm pretty sure the solution is not supposed to be growing but I guess I could be wrong.

Any help with the DE is greatly appreciated!


2 Answers 2


The reason Mathematica says it got division by zero, because your initial conditions are not valid. They produce no solution.

The analytical solution to

$$ \left( 1+\sqrt {\pi\,x} \right) y \left( x \right) +x{\frac {\rm d}{ {\rm d}x}}y \left( x \right) -1/6\,{x}^{2}{\frac {{\rm d}^{2}}{{\rm d} {x}^{2}}}y \left( x \right) =0 $$

Is given by Maple as (DSolve does not seem to be able to solve this, after some time waiting).

$$ y \left( x \right) ={\it \_C1}\,{x}^{7/2+1/2\,\sqrt {73}} {\mbox{$_0$F$_1$}(\ ;\,1+2\,\sqrt {73};\,24\,\sqrt {\pi}\sqrt {x})}+{ \it \_C2}\,{x}^{7/2-1/2\,\sqrt {73}} {\mbox{$_0$F$_1$}(\ ;\,1-2\,\sqrt {73};\,24\,\sqrt {\pi}\sqrt {x})} $$

Where $F_1$ is hypergeom function. Taking derivative of the above gives

$$ {\frac {\rm d}{{\rm d}x}}y \left( x \right) ={\frac {{\it \_C1}\,{x}^{ 7/2+1/2\,\sqrt {73}} \left( 7/2+1/2\,\sqrt {73} \right) {\mbox{$_0$F$_1$}(\ ;\,1+2\,\sqrt {73};\,24\,\sqrt {\pi}\sqrt {x})}}{x }}+12\,{\frac {{\it \_C1}\,{x}^{7/2+1/2\,\sqrt {73}} {\mbox{$_0$F$_1$}(\ ;\,2+2\,\sqrt {73};\,24\,\sqrt {\pi}\sqrt {x})} \sqrt {\pi}}{ \left( 1+2\,\sqrt {73} \right) \sqrt {x}}}+{\frac {{\it \_C2}\,{x}^{7/2-1/2\,\sqrt {73}} \left( 7/2-1/2\,\sqrt {73} \right) {\mbox{$_0$F$_1$}(\ ;\,1-2\,\sqrt {73};\,24\,\sqrt {\pi}\sqrt {x})}}{x }}+12\,{\frac {{\it \_C2}\,{x}^{7/2-1/2\,\sqrt {73}} {\mbox{$_0$F$_1$}(\ ;\,2-2\,\sqrt {73};\,24\,\sqrt {\pi}\sqrt {x})} \sqrt {\pi}}{ \left( 1-2\,\sqrt {73} \right) \sqrt {x}}} $$

You can now see that at $x=0$ there is a division by zero.

So your initial conditions are not valid or something wrong with your equations (may be did not copy it correctly from the book?).

  • $\begingroup$ What a formidable solution! I see this now. Thank you, I'll mark this as correct. $\endgroup$
    – Buddhapus
    Jun 17, 2017 at 6:57

The easy way out is to give a starting value to $x$ to be 10^-4 not 0.

NDSolve[{(1 + Sqrt[π x])*y[x] + x*y'[x] - (x^2)/6*y''[x] == 1, y[10^-4] == 1, 
          y'[10^-4] == 0}, y[x], {x, 10^-4, 2}]

Plot[y[x] /. %, {x, 10^-4, 2}]
  • $\begingroup$ I was able to find this solution before but I'm not sure it's physical. I'm looking for a decaying solution. $\endgroup$
    – Buddhapus
    Jun 16, 2017 at 9:33
  • $\begingroup$ @Buddhapus Then you should check your equation. $\endgroup$
    – zhk
    Jun 16, 2017 at 11:02

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