# Differential equation solution

DSolve[y''[x] + (a - (b*x^2 - c)*x^2)*y[x] == 0, y[x], x]


• There is no solution for the anharmonic oscillator in terms of the elementary or any standard special function as far as I know. – Nikolay Gromov Sep 17 '15 at 12:13
• I get a solution in terms of DifferentialRoot. What version are using? – Michael E2 Sep 17 '15 at 13:31

Mathematica can't solve analytically,but numerical yes can.

Clear["Global*"]
a = 3;
b = 2;
c = 1;
eq = {y''[x] + (a - (b*x^2 - c)*x^2)*y[x] == 0, y == 0, y' == 1};
sol = First@NDSolve[eq, y, {x, -2, 2}];
Plot[Evaluate[y[x] /. sol], {x, -2, 2}, PlotRange -> All,
PlotLegends -> {"y(x)"}, AxesLabel -> {x, y[x]}] MAPLE analytical solution: y[x] ==
C*HT[(2^(1/3)*3^(2/3)*(4*a*b + c^2))/(8*b^(4/3)),
0, (-c*2^(1/3)*3^(1/3))/(2*b^(2/3)), (-2^(1/3)*3^(2/3)*b^(1/6)*x)/
3]*Exp[(x*(2*b*x^2 - 3*c))/(6*Sqrt[b])] +
C*HT[(2^(1/3)*3^(2/3)*(4*a*b + c^2))/(8*b^(4/3)),
0, (-c*2^(1/3)*3^(1/3))/(2*b^(2/3)), (2^(1/3)*3^(2/3)*b^(1/6)*x)/
3]*Exp[(-(b*x^2 - 3/2*c)*x)/(3*Sqrt[b])]
`

You can be expressed this equation by the series:

$$y(x)=c_1 \text{HT1} \exp \left(\frac{x \left(2 b x^2-3 c\right)}{6 \sqrt{b}}\right)+c_2 \text{HT2} \exp \left(-\frac{\left(b x^2-\frac{3 c}{2}\right) x}{3 \sqrt{b}}\right)$$

were HT1 and HT2 is: and References for Triconfluent Heun function:

1.Decarreau, A.; Dumont-Lepage, M.C.; Maroni, P.; Robert, A.; and Ronveaux, A. "Formes Canoniques de Equations confluentes de l'equation de Heun". Annales de la Societe Scientifique de Bruxelles. Vol. 92 I-II, (1978): 53-78.

2.Ronveaux, A. ed. Heun's Differential Equations. Oxford University Press, 1995.

3.Slavyanov, S.Y., and Lay, W. Special Functions, A Unified Theory Based on Singularities. Oxford Mathematical Monographs, 2000.